Abstract

This work investigates the importance of verification and validation (V&V) to achieve predictive scale-resolving simulations (SRSs) of turbulence, i.e., computations capable of resolving a fraction of the turbulent flow scales. Toward this end, we propose a novel but simple V&V strategy based on grid and physical resolution refinement studies that can be used even when the exact initial flow conditions are unknown, or reference data are unavailable. This is particularly relevant for transient and transitional flow problems, as well as for the improvement of turbulence models. We start by presenting a literature survey of results obtained with distinct SRS models for flows past circular cylinders (CCs). It confirms the importance of V&V by illustrating a large variability of results, which is independent of the selected mathematical model and Reynolds number. The proposed V&V strategy is then used on three representative problems of practical interest. The results illustrate that it is possible to conduct reliable V&V exercises with SRS models and evidence the importance of V&V to predictive SRS of turbulence. Most notably, the data also confirm the advantages and potential of the proposed V&V strategy: separate assessment of numerical and modeling errors, enhanced flow physics analysis, identification of key flow phenomena, and ability to operate when the exact flow conditions are unknown or reference data are unavailable.

1 Introduction

Scale-resolving simulation (SRS) models are nowadays widely applied in computation of turbulent flow problems of practical interest, e.g., vehicle aerodynamics and hydrodynamics [17], offshore engineering [813], propeller design [1419], materials mixing [2025], and combustion [2631]. These formulations are characterized by their ability to resolve a fraction of or all turbulence scales present in a flow. This property unleashes the potential of SRS methods to obtain high-fidelity predictions of complex flows not amenable to modeling with the Reynolds-averaged Navier–Stokes (RANS) equations due to their limitations representing transient phenomena, onset and development of turbulence, instabilities, and coherent structures. Among all SRS formulations, hybrid [3237], bridging [3844], and large-eddy simulation (LES) [4547] are engineering practitioners' most usual modeling strategies.

The further establishment of SRS methods in many areas of science and engineering is inevitably dependent on the ability to obtain predictive computations, i.e., simulations with a quantified and acceptable level of numerical, input, and modeling uncertainty [48,49]. Yet, the estimation of these sources of computational uncertainty is difficult in SRS and, as such, often neglected. There are four primary reasons for this:

  • Complexity—SRS models calculate instantaneous variables which are closely dependent on the physical resolution or range of scales resolved by the model. This hampers the development of accurate and robust SRS turbulence closures by making the comparison and interpretation of some quantities difficult, as it is not possible to match the physical resolution of the reference experiments or simulations.

  • Cost—SRS calculations are usually computationally more intensive than RANS owing to the numerical requirements (spatiotemporal grid resolution, simulation time, iterative convergence criterion, etc.) necessary to accurately resolve a fraction of the turbulence field. This makes detailed verification and validation (V&V) exercises time consuming and resource intensive.

  • Dynamic modeling—The physical resolution of most SRS formulations is defined based on the grid resolution and the simulated flow properties to optimize the use of the available computational resources. However, such modeling option makes the formulation's physical resolution vary upon grid refinement. This turns SRS models grid dependent, precluding the evaluation of modeling and numerical errors separately, a requirement for reliable V&V exercises. Also, dynamic SRS formulations lead to commutation errors, since the filtering operator of the model does not commute with spatial and temporal differentiation [50].

  • Numerical setup—A precise validation exercise requires numerical simulations and reference experiments performed on the same flow conditions, i.e., domain, boundary conditions, material properties, and initial or inflow conditions. For SRS, it is not enough to know the mean values of these quantities; initializing the model requires both the time-resolved large-scale motions and the detailed statistics of the small scales. However, there are many cases where some of this information is not available from the reference experiments. This is particularly important for transient and transitional flows and may significantly affect proper comparisons between simulations and reference experiments.

To illustrate the role of V&V exercises [48,5153] to numerical prediction and turbulence modeling, Fig. 1 depicts the outcome of a literature survey [49,54] that compiles results for the time-averaged drag coefficient obtained for the flow around a circular cylinder (CC) at different Reynolds numbers, Re. One set of experimental measurements is included for comparison [55], and the numerical results are colored by category of model: direct numerical simulation (DNS), implicit large-eddy simulation (ILES), LES, the bridging partially averaged Navier–Stokes (PANS) equations, hybrid, and RANS. Note that except for RANS, all these formulations are SRS models. The numerical results exhibit an enormous range of values, which does not appear to improve for any specific mathematical model. Nor does it converge with physical resolution, from RANS where turbulence is completely modeled, to LES/ILES where most turbulent scales are resolved. Considering the case Re=3900, the collected C¯D results vary from 0.6 to 1.8, representing a maximum variation of 191% (taking C¯D=0.6 as the reference). Overall, Fig. 1 emphasizes the need for V&V exercises to further establish SRS methods in engineering and generate confidence in their numerical results when reference solutions are unavailable.

Fig. 1
Literature survey of numerical results for the time-averaged drag coefficient, C¯D, for the flow around a circular cylinder at different Re [49,54]
Fig. 1
Literature survey of numerical results for the time-averaged drag coefficient, C¯D, for the flow around a circular cylinder at different Re [49,54]
Close modal

This study investigates the importance of V&V exercises to achieve predictive SRS of turbulence. This is a necessary step to diminish the variability of SRS results in Fig. 1 and gain confidence on such formulations. Toward this end, we propose a novel but simple V&V approach based on grid and physical resolution refinement studies that can be used even when the initial flow conditions are not sufficiently characterized, or reference results are unavailable. This is crucial for transient and transitional flow simulations and for the improvement of the accuracy of turbulence models. The proposed method is utilized to ascertain the simulations' fidelity of three representative test-cases involving transition to turbulence: the flow around a CC at Re=3.9×103, the Taylor–Green vortex (TGV) at Re=3000 [56], and the Rayleigh–Taylor (RT) flow [57,58] at Atwood number At=0.5. The first problem is a statistically unsteady flow with well-characterized flow conditions and available experimental results and uncertainties, enabling the estimation of validation uncertainties. The second test-case is a transient flow with exact flow conditions and available reference DNS results. Yet, the numerical uncertainty of these DNS results has not been estimated. The third problem is a transient flow without reference experimental results due to the complexity of characterizing the exact experimental RT flow conditions (see literature survey of Pereira et al. [59]). These three test-cases constitute the ideal validation space for the new V&V method. All computations are based on the PANS equations [23,42,60] at constant physical resolution to enable the assessment of numerical and modeling errors separately and prevent commutation errors.

This paper is structured as follows. Section 2 describes the proposed V&V strategy, and Sec. 3 presents the details of the selected test-cases and numerical simulations. Afterward, the results are discussed in Sec. 4, and the conclusions are summarized in Sec. 5.

2 Verification and Validation Strategy

Before describing the proposed V&V strategy, let us start by introducing the concepts of error and uncertainty. Following the American Society of Mechanical Engineers standard for V&V [48,52], an error can be defined as the difference between the observed, ϕ, and the exact or truth, ϕo, solution
E(ϕ)ϕϕo
(1)
where ϕo depends on the class of errors being quantified. Yet, exact solutions are usually unavailable for most engineering flows and so we estimate the uncertainty of ϕ,U(ϕ), that establishes an interval that should contain the exact solution with a given degree of confidence. In the American Society of Mechanical Engineers V&V 20 standard [48], uncertainty is defined with a ± sign to obtain an interval
ϕU(ϕ)ϕoϕ+U(ϕ)
(2)

Therefore, unlike an error, uncertainty is always a positive quantity.

The uncertainty is evidently related to the error of a simulation, Et(ϕ), that includes three components
Et(ϕ)Ei(ϕ)+Em(ϕ)+En(ϕ)
(3)

Here, Ei(ϕ) is the input or parameter error caused by inexact values of boundary conditions, material properties, or initial conditions, Em(ϕ) is the modeling error defined as the difference between the exact solution of the mathematical model and physical truth, and En(ϕ) is the numerical error which can be divided into a discretization, iterative, round-off, and statistical (unsteady flows like the CC case) component [48,5153].

The goal of a validation assessment is to estimate Em(ϕ). However, as illustrated by Eq. (3), the estimation of Em(ϕ) depends on input, Ei(ϕ), and numerical, En(ϕ), errors. Furthermore, the true physical value is obtained from an experiment that also includes an experimental error Ee(ϕ). Therefore, the estimation of the modeling error Em(ϕ) is performed with an interval centered at the difference between the simulation ϕ and the experimental data ϕe (comparison error, Ec(ϕ)=ϕϕe) and a width defined by the so-called validation uncertainty, Uv(ϕ) [48],
Ec(ϕ)Uv(ϕ)Em(ϕ)Ec(ϕ)+Uv(ϕ)
(4)
Uv(ϕ) depends on the input, Ui(ϕ), numerical, Un(ϕ), and experimental, Ue(ϕ), uncertainties. For cases where these three uncertainties are independent, we obtain
Uv(ϕ)2=Ui(ϕ)2+Un(ϕ)2+Ue(ϕ)2
(5)

The estimation of Uv(ϕ) can be quite challenging for problems in which the initial flow conditions are not well characterized (large Ui(ϕ)). Also, the calculation of Uv(ϕ) is not possible in problems where reference experiments are unavailable. This is often the case for such problems as transitional flow, high-speed combustion, climate prediction, and full-scale problems. In addition to these difficulties, the governing equations of most SRS models directly depend on the grid resolution. This feature prevents the separate and reliable quantification of numerical and modeling errors (En(ϕ) and Em(ϕ)) until all flow scales are resolved in a DNS.

Therefore, the numerical prediction of complex flows with SRS models raises the following question: how can one conduct reliable V&V exercises to evaluate the accuracy of the computations? We propose the following strategy. Let us start by considering an SRS formulation whose governing equations do not depend on the grid resolution. This requires a model that uses an explicit type of filtering operator. Hence, one can evaluate modeling and numerical errors separately, and the conditions necessary to derive SRS models and assure scale-invariance [61] are satisfied. It also prevents commutation errors [50]. In this work, we use the PANS equations [42] supplemented with turbulence closures for incompressible single fluid [60,62] and variable-density flow [23]. We emphasize that we could have selected other SRS formulations as long as they use an explicit closure to represent the unresolved flow scales that can operate at constant physical resolution. This might require model modifications. For example, traditional SRS models utilizing a local characteristic grid size ΔSRS to set the physical resolution (e.g., detached eddy simulation (DES), very large simulation (VLES), extra large eddy simulation (XLES), LES, etc.) can be modified to either use a uniform ΔSRS or the ΔSRS distribution of the coarsest grid used in the study. This strategy removes the governing equations dependence from the grid cell size, is simple to implement, and enables decoupling numerical from modeling errors. Alternatively, one can use explicit filtering. Yet, the first approach is better suited for practical simulations of turbulence.

In the PANS models of Pereira et al. [23,60,62], the physical resolution is set constant, and it is defined by the ratio of the unresolved to the total turbulence kinetic energy
fk=kukt
(6)

where ku is the unresolved or modeled turbulence kinetic energy, and kt is the total. Note that fk = 0 is equivalent to DNS because all turbulence scales are resolved, whereas fk = 1 is equivalent to RANS since the turbulence closure models all turbulence scales. For a better physical understanding of the impact of fk on the predicted flow fields, Fig. 2 illustrates how the computed velocity field evolves with the relative filter length size, n. n equal to one is a direct numerical simulation (fk = 0) where all flow scales are resolved, leading to a highly unsteady velocity field with steep gradients that increase the cost of numerical simulations substantially. In contrast, n = 349 models most of the turbulence field (fk0.96), which decreases the cost of the computations.

Fig. 2
Evolution of x1-component of the forced homogeneous isotropic turbulence flow [63] velocity field with the relative filter length size, n=Δ/Δη [23,49]. Δη is the grid size of the DNS simulation. (a) n = 1, (b) n = 17, (c) n = 33, (d) n = 69, (e) n = 199, and (f) n = 349.
Fig. 2
Evolution of x1-component of the forced homogeneous isotropic turbulence flow [63] velocity field with the relative filter length size, n=Δ/Δη [23,49]. Δη is the grid size of the DNS simulation. (a) n = 1, (b) n = 17, (c) n = 33, (d) n = 69, (e) n = 199, and (f) n = 349.
Close modal

This class of SRS formulations provides the background to propose the new V&V strategy based on grid and physical resolution refinement exercises. It enables the evaluation of the numerical accuracy of the simulations through grid refinement exercises performed at constant fk, and the assessment of the modeling accuracy of the calculations through physical resolution refinement studies since the solutions converge toward the “truth” as fk0. Yet, the solutions are expected to converge at coarser physical resolutions than the limiting case of fk = 0 (DNS for a continuous medium). This is shown below and in numerous studies [12,59,60,6468], pointing out the method's potential to be applied to complex new problems without experimental solutions. The latter feature is a unique feature of the new V&V method, which can be crucial to many complex flow problems.

Figure 3 illustrates this strategy's typical outcome by depicting the evolution of a generic quantity ϕ with the grid refinement ratio
ri=N1Ni3
(7)

and/or the physical resolution, fk. In relation (7), N1 and Ni are the number of cells of the finest grid available (i = 1) and a selected grid i. The data can be assessed and visualized in three ways: (i) fixing fk and plotting ϕ(ri); (ii) fixing ri and depicting ϕ(fk); and (iii) plotting ϕ(fk,ri). This last approach provides a more comprehensive analysis, since it enables the assessment of the numerical and modeling accuracy, their interaction and interdependency, and the interpretation of the predicted flow physics and its dependence on fk (see Sec. 4). Figure 3 shows that the solutions of ϕ converge both with the grid and physical resolutions as ri0 and fk0. Note that the smallest ri shown throughout this work is equal to one, corresponding to the solution obtained on the finest grid tested. ri = 0 would require the extrapolation of the solution to an infinitely fine grid (e.g., Refs. [51] and [69]). From the interpretation of Fig. 3, it is possible to infer that the present technique can be utilized when the experimental flow conditions are poorly characterized or unknown or in cases in which the reference solution is unavailable. In such cases, one can use the extrapolated solution (ri0 and fk0) as reference. It is important to mention that the former procedure could be “generalized” to cases of grid dependent SRS models by considering the cross-derivatives with respect to ri and fk, (2ϕ)/(rifk).

Fig. 3
Sketches of generic quantity, ϕ, evolution upon grid, ri, and physical, fk, resolution refinement: (a) ϕ(ri) or ϕ(fk) and (b) ϕ(ri, fk)
Fig. 3
Sketches of generic quantity, ϕ, evolution upon grid, ri, and physical, fk, resolution refinement: (a) ϕ(ri) or ϕ(fk) and (b) ϕ(ri, fk)
Close modal

The proposed strategy can evidently be combined with other V&V techniques [5153,6978]. In this work, we estimate the numerical uncertainty of the simulations through the method proposed by Eça and Hoekstra [69], and its simplified version utilized in Ref. [60] (CC flow).

3 Test-Cases and Numerical Details

3.1 Flow Problems.

We use three representative flow problems to investigate the importance of V&V to SRS of turbulence and evaluate the potential of the proposed V&V strategy. The first problem is a statistically unsteady flow with well-characterized flow conditions and available experimental results and uncertainties, enabling the estimation of validation uncertainties. The second test-case is a transient flow with exact flow conditions and available reference DNS results. Yet, the numerical uncertainty of these DNS results has not been estimated. The third problem is a transient flow without reference experimental results due to the complexity of characterizing the exact experimental initial flow conditions. These three test-cases constitute the ideal validation space for the new V&V method and are described below.

3.1.1 Circular Cylinder.

The flow past a CC at Re=3900 is an archetypal problem commonly used to assess the prediction of transition, turbulence, and separation around bluff-flows in the subcritical regime [79,80]. This problem is typical of model-scale applications and unmanned vehicles. The flow is statistically unsteady and characterized by a laminar boundary-layer, flow separation leading to a free shear-layer, onset and development of turbulence in the free shear-layer, and finally a turbulent wake. Also, the flow physics is closely dependent on two coherent structures: the Kelvin–Helmholtz in the free shear-layer and the large-scale vortex-shedding in the wake. The first is responsible for transition to turbulence in the free shear-layer. The spatial development of the flow in this regime comprises four major steps, proper representation of which determines the accuracy of any numerical simulation: (i) onset of the Kelvin–Helmholtz instability in the free shear-layer; (ii) spatial development of the Kelvin–Helmholtz rollers; (iii) breakdown to high-intensity turbulence; and (iv) turbulent free shear-layer roll-up, leading to vortex-shedding. A comprehensive numerical analysis of this problem is given in Refs. [60], [64], and [65].

In this work, we use the experimental measurements of Parnaudeau et al. [81] as reference to evaluate the modeling error. These have been conducted in a wind-tunnel with a square cross section of width equal to 23.3D (D is the cylinder's diameter), and a uniform inflow characterized by a turbulence intensity smaller than I=0.2%. The Reynolds number, ReVoD/ν, is equal to 3900 (Vo is the inflow velocity, and ν is the kinematic viscosity), and the sampling period used to compute the flow statistics is ΔTVo/D=2.08×105. The quantity of interest extracted from this study is the time-averaged velocity field, V¯i. This has been measured using particle image velocimetry, and the reported experimental uncertainties do not exceed 1.0%. We also used the experiments of Norberg [82,83] to assess the predictions of the time-averaged drag coefficient, C¯D, root-mean-square lift coefficient, CL, and pressure coefficient on the cylinder's surface, C¯p(θ). Compared to Parnaudeau et al. [81], these experimental studies are conducted with two slightly different setups: Re=4000,215.5D×80.0D cross section, I<0.06%, and ΔTVo/D=2.10×104 for Ref. [82], whereas Re=4400,314.1D×105.0D cross section, and ΔTVo/D=3.20×104 for Ref. [83]. The reported experimental uncertainties are 0.01 for C¯p(θ), 1.0% for C¯D, and 4.0% for CL.

The numerical simulations try to replicate the experimental apparatus of Parnaudeau et al. [81] closely. The computational domain is shown in Fig. 4(a). The domain is 50D long, with the inlet at 10D from the inflow, 24D wide, and 3D thick, with cylinder axis centered. The impact of the cylinder's length (L=3D) has been ascertained and guarantees acceptable input uncertainties [54]. At the inlet (x1/D=10), the prescribed boundary conditions are constant velocity (Vo) aligned with the streamwise direction x1, and the pressure is extrapolated from the interior of the domain. The turbulence quantities are set to result in a turbulence intensity I=0.2% and a ratio between turbulent and molecular kinematic viscosity equal to 103. At the outlet (x1/D=40), all streamwise derivatives are set equal to zero, whereas no-slip and impermeability conditions are applied on the cylinder's surface. Also, the normal pressure gradient and the turbulence quantities are equal to zero. The exception is the specific dissipation which is given at the center of the nearest-wall cell by ω=80νd2 (d is the nearest-wall distance). The pressure and transverse derivatives of the remaining dependent quantities are equal to zero at x2/D=±12, and symmetry conditions are prescribed at x3/D=0 and 3. The simulations are initialized from a 200 time units (ΔTVo/D) computation and run for a minimum of 500 time units. The flow statistics are calculated using a sampling period ΔTVo/D350 [49].

Fig. 4
Sketch of the computational domain and initial flow conditions of the CC, TGV, and RT problems: (a) CC, (b) TGV, and (c) RT
Fig. 4
Sketch of the computational domain and initial flow conditions of the CC, TGV, and RT problems: (a) CC, (b) TGV, and (c) RT
Close modal

3.1.2 Taylor–Green Vortex.

The TGV [56] is a benchmark transient problem used to investigate the modeling and simulation of transition to turbulence driven by vortex stretching and reconnection mechanisms [84]. The TGV problem starts with a laminar, single Fourier mode, well-defined vortex in a periodic box, illustrated in Fig. 4(b). This is given mathematically by [56,84]
V1(x,to)=Vosin(x1)cos(x2)cos(x3)
(8)
V2(x,to)=Vocos(x1)sin(x2)cos(x3)
(9)
V3(x,to)=0
(10)
where Vo is the initial velocity magnitude. The corresponding pressure field is obtained from solving the Poisson equation
P(x,to)=Po+ρoVo216[2+cos(2x3)][cos(2x1)+cos(2x2)]
(11)

Here, Po and ρo are the pressure and density magnitude at t = 0. After t = 0, these structures interact and deform, leading to the formation of vortex sheets. Afterward, the structures get closer and roll-up, causing the onset of turbulence due to a complex vortex reconnection process between pairs of counter-rotating vortices [68]. In the next instants, the remaining vortical structures breakdown, and turbulence further develops. This leads to rapid dissipation of the flow kinetic energy. Since the TGV is an isolated system, the flow kinetic energy decays in time.

The flow configuration of the simulations replicates that used in the DNS studies of Brachet et al. [84] and Drikakis et al. [85] at ReVoLo/ν=3000. The simulations are conducted in a cubical computational domain with width L=2πLo, where periodic conditions are applied on all boundaries. The initial Mach number is set Mao=0.28, and the simulations run until t=20Vo/Lo. The initial thermodynamic and flow properties are the following: vortex velocity Vo=104 cm/s, box size Lo=1.00 cm, density ρo=1.178×103 g/cm3, mean pressure Po=105 Pa, dynamic viscosity μ=3.927×103 g/(cm s), heat capacity ratio γ=1.40, turbulence kinetic energy ko=107 cm2/s2, and dissipation length scale So=6.136×103 cm. The DNS studies of Brachet et al. [84] at Ma=0 (incompressible) and Drikakis et al. [85] at Mao=0.28 are used to evaluate the modeling accuracy of the simulations. These do not report numerical uncertainties. We emphasize that the differences in Mao between our simulations and Brachet et al. [84] computations may lead to discrepancies.

3.1.3 Rayleigh–Taylor.

The RT [57,58] is a benchmark test-case of variable-density flow. This problem is a fundamental test-case for flows with material mixing and turbulence generation by baroclinicity. The flow initially consists of two fluids of different densities separated by a perturbed interface. These materials are at rest, with the dense medium (ρh) located on top of the light material (ρl) as shown in Fig. 4(c). Immediately after t = 0, the heavy fluid accelerates downward by the action of a body force, whereas the light fluid moves upwards. The interface perturbations create a misalignment between the density gradient and the pressure, which induces the RT instability. This generates structures: penetration of heavy fluid into the light fluid (called bubbles) and, conversely, spikes by penetration of light fluid into the heavy fluid. The boundaries of these structures experience shearing, resulting in a Kelvin–Helmholtz instability, which triggers the onset of turbulence. Afterward, turbulence further develops, enhancing the mixing of the two materials. The density difference is characterized by the Atwood number, At=(ρhρl)/(ρh+ρl).

The present flow configuration is the same as that of Pereira et al. [23,59] at At = 0.50. The computational domain is a rectangular prism with a squared cross section of width Lo=2πcm and a height of 3Lo. The mixing-layer height does not exceed 2Lo during a simulation time of 20 time units (time normalized by t*=Lo/(32gAt) [86]). Periodic conditions are applied on the lateral boundaries, whereas reflective conditions are prescribed on the top and bottom boundaries. The interface is perturbed as in Ref. [59] with density fluctuations, which height is defined by
hp(x1,x3)=n,mcos[2π(nx1Lo+r1)]cos[2π(mx3Lo+r3)]
(12)

The perturbations' wavelengths range between modes 30 and 34 (30n2+m234), and their amplitude is featured by a maximum standard deviation equal to 0.04Lo. In Eq. (12), the modes m and n are selected to include the most unstable mode of the linearized problem, and r1 and r3 are random numbers between 0 and 1. The present numerical experiments use the ideal gas equation of state, and the initial temperature is set to maintain the flow Ma<0.10. The initial thermodynamic and flow properties are defined as follows: μl=0.002 g/(cm s), μh=0.006 g/(cm s), ρl=1.0 g/cm3, ρh=3.0 g/cm3, γl=γh=1.40, g = −980 cm/s2, ko=106cm2/s2,So=106cm, and Schmidt and Prandtl numbers equal to one.

3.2 Numerical Settings.

The CC flow is calculated with the incompressible flow solver refresco [87]. This is a community-based open-usage flow solver optimized for maritime applications. refresco solves multiphase incompressible viscous flows using the filtered continuity and Navier–Stokes equations, complemented with turbulence closures. The equations are discretized using a finite volume approach with cell-centered collocated variables, in strong-conservation form, and a pressure-correction equation based on the semi-implicit method for pressure linked equations (SIMPLE) algorithm is used to ensure mass conservation. Time integration is performed implicitly with a second-order backward scheme. The spatial discretization schemes are also second-order accurate, and we use a second-order upwind-based scheme to discretize the convective terms of the governing equations. Four spatiotemporal grid resolutions are used, ranging from 1.04×106 to 4.55×106 cells and time-steps between 8.53×103 and 5.21×103 time units. These lead to a refinement ratio r of approximately 1.64. At each implicit time-step, the nonlinear system for velocity and pressure is linearized with Picard's method, and a segregated approach is adopted for the solution of all transport equations. The simulations run on double precision, and the iterative convergence criterion [49] guarantees that the maximum normalized residual of all dependent quantities is always smaller than 105.

The implementation is face-based, which permits grids with elements consisting of an arbitrary number of faces. refresco was initially developed as a RANS solver [88], where the Reynolds-stress tensor was modeled through linear turbulent viscosity closures. Pereira and co-workers [54,89] extended the available mathematical models and turbulence closures to PANS, XLES, delayed DES (DDES), improved DDES (IDDES), RANS-explicit algebraic Reynolds stresses model (EARSM), and RANS-Reynolds stresses model (RSM) methods. The code is parallelized using message passing interface (MPI) and subdomain decomposition and runs on Linux workstations and high performance computing (HPC) clusters. The code is currently being developed, verified, and tested at MARIN (the Netherlands) in collaboration with several universities [87].

The TGV and RT calculations are conducted with the compressible flow solver xrage [90]. This code utilizes a finite volume approach to solve the compressible and multimaterial conservation equations for mass, momentum, energy, and species concentration. The resulting system of governing equations is resolved through the Harten–Lax–van Leer-contact [91] Riemann solver using a directionally unsplit strategy, direct remap, parabolic reconstruction [92], and the low Mach number correction proposed by Thornber et al. [93]. The equations are discretized with second-order accurate methods: the spatial discretization is based on a Godunov scheme, whereas the temporal discretization relies on an explicit Runge–Kutta scheme known as Heun's method. The time-step, Δt, is defined by prescribing the maximum instantaneous Courant– Friedrichs–Lewy (CFL) number
Δt=ΔxCFL3(|V|+c)
(13)
where c is the speed of sound, and Δx is the grid size. The CFL number is set equal to 0.45 for the TGV and 0.50 for the RT. The code can utilize an adaptive mesh refinement algorithm for following waves, especially shock-waves and contact discontinuities, but this option is not used in this work to prevent hanging-nodes [94] and, as such, the simulations use orthogonal uniform hexahedral grids. These have 1283, 2563, and 5123 (and 10243 for fk=0.00) cells for the TGV and 642×192,1282×384,and2562×768 (and 5122×1536 with CFL =0.8 for fk=0.00) cells for the RT.xrage models miscible material interfaces and high convection-driven flows with a van Leer limiter [95], without artificial viscosity, and no material interface treatments [96,97]. The solver uses the assumption that cells containing more than one material are in pressure and temperature equilibrium as a mixed cell closure. The effective kinematic viscosity in multimaterial problems [24] is defined as
ν=n=1ntνnfn
(14)

where n is the material index, nt is the number of materials, and fn is the volume fraction of material n. For the RT flow, the diffusivity D and thermal conductivity κ are defined by imposing the Schmidt (Scν/D) and Prandtl (Prcpμ/κ) numbers equal to one.

4 Results

The importance of V&V to SRS and the advantages of the proposed strategy are illustrated in this section. This is accomplished through its application to the results of the circular cylinder [60,64,65], Taylor–Green vortex [68], and Rayleigh–Taylor [23,59].

4.1 Circular Cylinder.

We initiate the discussion of the results with the flow past a circular cylinder in the subcritical regime. Figure 5 presents change with grid, ri, and physical resolution, fk, refinement in three time-averaged quantities: the drag coefficient, C¯D, base pressure coefficient, C¯pb, and recirculation length, L¯r, and one root-mean-square quantity, RMS lift coefficient, CL. As expected, the simulations converge both with numerical resolution (ri0) and physical resolution (fk0). The results approach the experimental values with varying fidelity depending on the quantity. Yet, it is still possible to observe small discrepancies between the converged numerical results and the experiments. These discrepancies are caused by differences between the numerical and experimental apparatus, i.e., input errors. The experimental flow at the inlet is difficult to replicate because the spectral properties of the turbulence at the inlet were not experimentally measured (only the averaged quantities are available). The length and times scales of disturbances are crucial for transitional problems and particularly important for numerical predictions of the recirculation region length [98101]. The differences in aspect ratio (length) of the cylinder [102104], surface roughness [98,105,106], or Re [55] can also have a minor impact to the differences between numerical and experimental measurements. These results reaffirm the importance of the input error and also show the potential of ri and fk refinement studies to create reference solutions that can be used to analyze complex flows. Without all the information in Fig. 5, one could argue that an SRS model at fk=0.40 is “better” calculating L¯r than formulations at finer physical resolutions and draw modeling conclusions based on such idea. However, such a conclusion would be the result of insufficient data and not properly identifying error cancellation between input and modeling/numerical errors. In this case, the input error decreases L¯r, while the modeling/numerical error increases it.

Fig. 5
Evolution of C¯D, CL′, C¯pb, and L¯r upon grid, ri, and physical, fk, resolution refinement: (a) C¯D, (b) CL′, (c) C¯pb, and (d) L¯r
Fig. 5
Evolution of C¯D, CL′, C¯pb, and L¯r upon grid, ri, and physical, fk, resolution refinement: (a) C¯D, (b) CL′, (c) C¯pb, and (d) L¯r
Close modal

Figure 5 also shows that the selected mathematical model can accurately predict the present flow problem provided an adequate value of fk and sufficient grid resolution. The results show only small discrepancies between experiments and simulations at fk<0.50. This is also seen in the values of Ec(ϕ) and Uv(ϕ) presented in Table 1 for three fk (see Refs. [60] and [65] for the comprehensive analysis). These quantities are obtained using the V&V20 metric [48]. For example, the results show that the values of CL and C¯pb decrease from 0.664 and 1.41 at fk=1.00 (RANS) to 0.095 and 0.86 at fk=0.25, where the experimental values are equal to 0.096 and 0.86, respectively. This constitutes a reduction of Ec(CL) from 591.8% to −1.1% and Ec(C¯pb) from 59.7% to −1.8%.

Table 1

Estimated comparison error, Ec(ϕ), and validation uncertainty, Uv(ϕ), for the time-averaged drag coefficient, C¯D, root-mean-square lift coefficient, CL, time-averaged base pressure coefficient, C¯pb, and recirculation length, L¯r, for different fk

ModelΦC¯DCLC¯pbL¯r
fk=1.00ϕ1.250.6641.410.51
Ec(ϕ)+27.1+591.8+59.7−63.8
Uv(ϕ)±1.5±57.6±16.2±49.7
fk=0.50ϕ1.040.2841.051.12
Ec(ϕ)+5.7+195.7+19.3−25.6
Uv(ϕ)±32.2±549.7±61.7±62.0
fk=0.25ϕ0.930.0950.861.73
Ec(ϕ)−5.4−1.1−1.8+14.4
Uv(ϕ)±3.9±55.9±6.4±20.2
Expt.ϕe0.980.0960.881.51
Ue±0.01±0.004±0.01±0.02
ModelΦC¯DCLC¯pbL¯r
fk=1.00ϕ1.250.6641.410.51
Ec(ϕ)+27.1+591.8+59.7−63.8
Uv(ϕ)±1.5±57.6±16.2±49.7
fk=0.50ϕ1.040.2841.051.12
Ec(ϕ)+5.7+195.7+19.3−25.6
Uv(ϕ)±32.2±549.7±61.7±62.0
fk=0.25ϕ0.930.0950.861.73
Ec(ϕ)−5.4−1.1−1.8+14.4
Uv(ϕ)±3.9±55.9±6.4±20.2
Expt.ϕe0.980.0960.881.51
Ue±0.01±0.004±0.01±0.02

Ec(ϕ) and Uv(ϕ) are shown as percentage of the experimental value [8183].

We need to emphasize that CL is a quantity that does not permit direct comparisons of simulations at different physical resolutions (consistent with the first challenge involved in V&V of SRS identified in Sec. 1). Its magnitude will increase upon fk refinement, as more of the fluctuations which contribute to CL are resolved, and this behavior is independent of the modeling accuracy of the simulation. The present results show that the modeling error of simulations at fk=1.00 is larger than this effect since CL grows as fk1.00.

A very significant result of Fig. 5 from a V&V perspective is the minima observed on the solutions convergence with fk on the two coarsest grids (ri1.35). This behavior occurs at fk0.50 and leads to significantly larger comparison errors at the finest physical resolution (fk=0.15) than at 0.25. It is caused by insufficient grid resolution to resolve all scales at fk=0.15 and demonstrates that the advantages and potential of SRS models can only be achieved with sufficient numerical resolution. The proposed V&V strategy identifies this behavior because it clearly distinguishes between the numerical errors and the modeling errors. Importantly, the method does not require reference data to make this distinction.

Another interesting result in Fig. 5 is the considerable dependence of solutions near fk=0.50 on the grid resolution (and fk). As observed in Table 1, the validation uncertainty is highest at fk=0.50. This can be explained in terms of the sensitivity of the flow to the effective Reynolds number, Ree,
ReeVoDνe
(15)
where νe is an effective viscosity, which can be decomposed as
νe=ν+νt+νn
(16)

with ν, νt, and νn the molecular, turbulent, and numerical kinematic viscosities, respectively. Note that νn comes from the numerical error. As demonstrated in Ref. [64], Ree can determine the ability of a mathematical model to capture the key instabilities and coherent structures driving this flow. Since fk dictates νt (and νn indirectly), simulations at values of fk close to the threshold at which the model begins to resolve the key instabilities and coherent structures of the flow are highly sensitive to νn. Alternatively, near a critical physical Reynolds number, the correct physical behavior is recovered only if both the turbulent and numerical viscosities are sufficiently small. Once again, this can be identified by the proposed V&V strategy. These results also represent crucial information for the utilization and development of SRS turbulence models, enabling a better understanding of the predicted flow physics and phenomena which are difficult to model. Finally, it is essential to mention that simulations at fk=0.75 can lead to slightly larger comparison errors than those at fk=1.00. This is a result of the calibration issues discussed in Sec. 1 that are not overcome by fk at coarse physical resolutions [11].

The trends of Fig. 5 and Table 1 are also observed in predictions of local flow quantities. This is shown in the following figures, which show the time-averaged pressure coefficient on the cylinder's surface, C¯p(θ), and the streamwise velocity profiles, V¯1, in the near-wake at different grid resolutions for constant fk=0.25 (Fig. 6) and different values of fk using the finest grid resolution (Fig. 7). All solutions converge upon the grid and physical resolution refinement and tend to the experimental measurement. Once again, the differences between simulations at different physical resolutions on the finest grid depend on the ability of the model to accurately resolve the instabilities driving the flow physics at a given fk. Such behavior is also observed for other SRS formulations [11,65].

Fig. 6
Evolution of C¯p(θ) and V¯1 upon grid refinement at fk=0.25: (a) C¯p(θ) and (b) V¯1
Fig. 6
Evolution of C¯p(θ) and V¯1 upon grid refinement at fk=0.25: (a) C¯p(θ) and (b) V¯1
Close modal
Fig. 7
Evolution of C¯p(θ) and V¯1 upon physical resolution refinement on grid g1: (a) C¯p(θ) and (b) V¯1
Fig. 7
Evolution of C¯p(θ) and V¯1 upon physical resolution refinement on grid g1: (a) C¯p(θ) and (b) V¯1
Close modal

4.2 Taylor–Green Vortex.

Next, we analyze the TGV results. This is a transient problem and, as such, V&V exercises are significantly more challenging and highly dependent on time. A flow misrepresentation at instant t will affect the subsequent instants and can compromise the overall simulation. This renders the accurate simulation and evaluation of the TGV (and RT) flow more complex than for the CC.

Figure 8 depicts the variation of the total (resolved and modeled) kinetic energy, k, with the grid, ri, and physical, fk, resolution at instants before, during, and after the onset of turbulence (the solution at fk = 0 on N=10243 is not included). As discussed in Ref. [68], the period t12 is crucial to the flow dynamics and simulation accuracy. At t = 5, all simulations converge upon grid refinement (ri0) and are independent of fk. This stems from the fact that the flow is laminar so that all simulations predict negligible magnitudes of turbulent stresses [68]. Later at t = 7, when the flow is supposed to undergo transition to turbulence, the computations become dependent on fk and ri. Yet, it is observed that the solutions converge upon the refinement of these parameters, and the discrepancies between simulations at fk0.35 are small. Also, it is possible to infer that varying physical and numerical resolution cause similar variations on the solutions of k. At later times, the impact of fk relative to ri grows significantly, but the solutions still converge upon fk and ri refinement.

Fig. 8
Temporal evolution of the total kinetic energy, k, with the grid and fk: (a) t = 5, (b) t = 7, (c) t = 9, and (d) t = 11
Fig. 8
Temporal evolution of the total kinetic energy, k, with the grid and fk: (a) t = 5, (b) t = 7, (c) t = 9, and (d) t = 11
Close modal

Figure 9 presents the temporal evolution of k and respective numerical uncertainty, Un(k), at different physical resolutions, fk. Un(k) is estimated with the method proposed by Eça and Hoekstra [69] using the three finest grids available: N=1283, 2563, and 5123 for all fk, and N=2563, 5123, and 10243 for fk=0.00. In Fig. 9, the black line indicates k(t), and the height of the red area the value of Un(k) at a given instant. We assume the discretization error as the main contributor to the numerical error. It is important to reiterate that Un(k) is the uncertainty associated with the numerical solution procedure, and so the red envelope in Fig. 9 is the range in which we expect the exact solution to the physical model to fall. The comparison error, Ec(k), which is the difference between the numerical solution and the reference solution, can be large or small, independent of the numerical uncertainty. For example, for fk = 1, the error is large (Fig. 8), because the model performs poorly, but the numerical uncertainty at late time can be small, if the numerical solution is well converged. This means the numerical solution accurately represents the mathematical model, but that the model does not accurately capture the physics. Conversely, for fk = 0, at late time, the modeling error is small, but the numerical uncertainty is expected to be larger (for the same grid) because of the high requirements for both numerical and statistical convergence of a fully resolved turbulent flow. For a better physical and numerical interpretation of the results, recall that the onset of turbulence is expected to occur at t7.

Fig. 9
Temporal evolution of the total kinetic energy numerical uncertainty, Un(k), for different fk: (a) fk = 1.00, (b) fk = 0.50, (c) fk = 0.35, (d) fk = 0.25, (e) fk = 0.00, and (f) fk = 0.00 on N = 10243
Fig. 9
Temporal evolution of the total kinetic energy numerical uncertainty, Un(k), for different fk: (a) fk = 1.00, (b) fk = 0.50, (c) fk = 0.35, (d) fk = 0.25, (e) fk = 0.00, and (f) fk = 0.00 on N = 10243
Close modal

Referring to the simulations at fk0.50, the results exhibit reduced Un(k) for most of the simulation time. Taking the case at fk=1.00 (RANS), the estimates of Un(k) do not exceed 3.3% of the predicted k. The exception lies in the period between t = 6 and 10, in which Un(k) can reach 10.8%. Such values of Un(k) are mostly caused by the difficulties of predicting the onset of turbulence with a RANS closure [68]. RANS models overpredict turbulence during transition, leading to a more rapid dissipation of k, which increases the flow gradients steepness and, consequently, the simulations' local numerical requisites. This illustrates the importance of a careful numerical and modeling interpretation of V&V exercises.

On the other hand, the refinement of fk to 0.35 and 0.25 reduces the predictions' numerical uncertainty significantly. Un(k) is negligible at early times and does not exceed 8.7% at late times when the flow features high-intensity turbulence and k is significantly smaller than at t = 0. However, the further refinement of physical resolution (fk=0.0) increases Un(k) at t > 11.0 considerably. This is the period when the flow is characterized by high-intensity turbulence and the largest numerical requirements since fk=0.00 aims to resolve all flow scales. For this reason, for simulations at fk=0.00, even the grid with N=10243 cells still has higher numerical uncertainties than those at fk=0.25 on N=5123. The simulation at fk=0.00 on the finest grid leads to (Un(k))max=8.4% at t = 13, whereas Un(k) does not exceed 2.9% at this instant for fk=0.25.

Now we turn our attention to Fig. 10, which presents the temporal evolution of k and its dissipation, ε,
ε=dkdt
(17)

obtained at different fk on the finest grid resolution available. The results of ε are compared to the DNS of Brachet et al. [84] (Ma=0,DNS1) and Drikakis et al. [85] (Mao=0.28,DNS2). These reference studies do not provide numerical uncertainties. Figure 10(a) shows that all simulations converge monotonically upon fk refinement, and those at fk0.35 exhibit small discrepancies, which do not exceed 0.0047. The exception in monotonic convergence is observed at t > 11 at fk0.35. This result is attributed to numerical uncertainty, especially for the case at fk=0.00 (Fig. 9). The results also show that all computations are independent of fk until ttc=6. This is the time when simulations at different fk lead to non-negligible levels of turbulence stresses.

Fig. 10
Temporal evolution of the total kinetic energy, k, and dissipation, ε, with different fk. DNS1, [84]; DNS2, [85]. (a) k and (b) ε.
Fig. 10
Temporal evolution of the total kinetic energy, k, and dissipation, ε, with different fk. DNS1, [84]; DNS2, [85]. (a) k and (b) ε.
Close modal

The results for ε depicted in Fig. 10(b) exhibit similar tendencies. Until ttc, all computations are independent of fk and lead to minor comparison errors. In contrast, the simulations become highly dependent on fk at t>tc, and their solutions converge upon fk refinement toward the reference data. Also, solutions of simulations at fk0.35 are significantly less dependent on fk, and the resulting comparison errors are small. Comparing the two DNS data sets, our results converge toward those of Drikakis et al. [85] (DNS2). This stems from the fact that our simulations match the Ma of such DNS, highlighting the role of input errors and Ma [107] to the comparisons. Regarding simulations at fk0.50, these lead to large comparison errors. For instance, the magnitude of the peak of dissipation predicted with fk=1.00 is 52.0% larger than that reported by Drikakis et al. [85].

In summary, Figs. 810 allow us to evaluate the simulations' modeling accuracy through ri and fk refinement studies. These converge toward the reference solutions, leading to small comparison errors. The results also reaffirm that the proposed V&V method can be confidently applied when reference data are unavailable or the flow conditions are unknown since it can generate a reference solution (ri0 and fk0). The interpretation of the results also indicates the existence of two groups of simulations: those at fk0.50 and fk<0.50. This can be explained by the critical value of fk that allows the SRS model to accurately resolve the key instabilities and coherent structures of the flow, not amenable to modeling. For TGV, these are the vortex reconnection phenomena responsible for the onset of turbulence [68].

4.3 Rayleigh–Taylor.

We conclude this section with a brief presentation of the RT flow, as these results exhibit behaviors similar to those of the CC and TGV. A complete discussion of the current results is given in Refs. [59] and [68]. The RT is a variable-density flow highly dependent on the initial conditions. Hence, we conduct simulations at different values of fk and compare the results against the case at fk = 0 to guarantee the same initial conditions for all values of fk [108]. Figure 11 presents the temporal evolution of the mixing-layer height, h, and the molecular mixing parameter, θ, predicted with different values of fk on the finest grid resolution available for all fk (N=2562×768). The second quantity quantifies the homogeneity of the mixing-layer, with θ = 0 and θ = 1 referring to a heterogeneous and homogeneous mixture, respectively.

Fig. 11
Temporal evolution of the mixing-layer height, h, and the molecular mixing parameter, θ, with different fk: (a) h and (b) θ
Fig. 11
Temporal evolution of the mixing-layer height, h, and the molecular mixing parameter, θ, with different fk: (a) h and (b) θ
Close modal

The results show that the simulations converge monotonically toward the reference solution upon physical resolution refinement, fk0. Also, whereas simulations at fk0.35 are in good agreement with the reference solution, those at fk0.50 lead to large discrepancies. As discussed in Refs. [23] and [59], this is caused by the ability of simulations at fk<0.50 to accurately predict the instabilities and coherent structures governing the flow dynamics. This requires resolving the RT instability, laminar spikes and bubbles, and the Kelvin–Helmholtz instability responsible for the onset of turbulence in the RT flow. Once again, the grid and fk refinement studies are critical to draw such conclusions.

Finally, Fig. 12 presents the numerical uncertainty of h obtained with fk = 0 (most demanding case) using the three finest grid resolutions—N=1282×384,2562×768, and 5122×1536 cells. The solutions of h for the three grids are also included. The results of Fig. 12(a) indicate that the numerical uncertainty of the simulations is relatively small and does not exceed 16.8% of the predicted value during the simulated time. Note that this represents an upper limit for Un(h), since the coarsest grid used to estimate the numerical uncertainty does not possess sufficient resolution to resolve the interface perturbations at t = 0. This idea is confirmed in Fig. 12(b), which compares the solutions of h for the three grids. Whereas the two finest grids lead to nearly identical solutions, the coarsest one leads to large discrepancies. Such outcome demonstrates the importance of (adequately resolving) the initial flow conditions, and the adequacy of the grid with 2562×768 cells for this study.

Fig. 12
Temporal evolution of the mixing-layer height, h, respective numerical uncertainty, Un(h), and grid resolution dependence for fk = 0: (a) h(t)±Un(t) and (b) h(N)
Fig. 12
Temporal evolution of the mixing-layer height, h, respective numerical uncertainty, Un(h), and grid resolution dependence for fk = 0: (a) h(t)±Un(t) and (b) h(N)
Close modal

5 Conclusions

This work investigated the importance of V&V to the credibility and further establishment of SRS of turbulence. A novel but simple V&V strategy based on grid and physical resolution studies is proposed that permits the separate assessment of numerical and modeling errors. It can be used when reference data are unavailable since it can estimate a reference solution from the physical resolution refinement studies. Also, it permits performing physical resolution refinements using the same initial flow conditions, avoiding the common difficulties generated by undercharacterized or unknown experimental conditions.

To ascertain the former aspects, we simulate three benchmark transitional flows using different grids and physical resolutions: the flow past a circular cylinder at Re=3900, the TGV at Re=3000, and the variable-density RT flow at At=0.50. The results confirm the importance of V&V to SRS of turbulence, indicate that it is possible to perform reliable V&V exercises with SRS formulations, and show the potential of the proposed V&V strategy. It enables an easier identification of cases experiencing error canceling, illustrates that insufficient grid resolution can suppress the modeling advantages of SRS of turbulence, permits a better flow analysis, and the identification of key flow phenomena. These properties can be used to develop better mathematical models and understand the reasons why models fail or succeed.

We expect the proposed V&V strategy to be a valuable contribution to further develop turbulence modeling, allow users attaining higher-quality computations, and promote the use of V&V in practical simulations of turbulence. Also, it can be a valuable tool for flows without available reference data or undercharacterized initial and flow experimental conditions. Naturally, performing grid and physical resolution refinement studies is computationally intensive, but the results of this study and Fig. 1 illustrate the risks of not assessing the accuracy of the simulations.

Acknowledgment

We would like to thank the reviewers for their suggestions that improved our paper. Los Alamos National Laboratory (LANL) is operated by TRIAD National Security, LLC for the U.S. DOE NNSA.

Funding Data

  • Laboratory Directed Research and Development program of Los Alamos National Laboratory (Project No. 20210289ER).

References

1.
Fureby
,
C.
,
Toxopeus
,
S. L.
,
Johansson
,
M.
,
Tormalm
,
M.
, and
Petterson
,
K.
,
2016
, “
A Computational Study of the Flow Around the KVLCC2 Model Hull at Straight Ahead Conditions and at Drift
,”
Ocean Eng.
,
118
, pp.
1
16
.10.1016/j.oceaneng.2016.03.029
2.
Nishikawa
,
T.
,
Yamade
,
Y.
,
Sakuma
,
M.
, and
Kato
,
C.
,
2012
, “
Application of Fully-Resolved Large Eddy Simulation to KCLCC2
,”
J. Jpn. Soc. Nav. Archit. Ocean Eng.
,
16
, pp.
1
9
.10.2534/JJASNAOE.16.1
3.
Krajnović
,
S.
, and
Davidson
,
L.
,
2005
, “
Flow Around a Simplified Car, Part 1: Large Eddy Simulation
,”
ASME J. Fluids Eng.
,
127
(
5
), pp.
907
918
.10.1115/1.1989371
4.
Minelli
,
G.
,
Krajnović
,
S.
, and
Basara
,
B.
,
2018
, “
A Flow Control Study of a Simplified, Oscillating Truck Cabin Using PANS
,”
ASME J. Fluids Eng.
,
140
(
12
), p.
121101
.10.1115/1.4040225
5.
Schauerhamer
,
D. G.
, and
Robinson
,
S. K.
,
2016
, “
A Validation Study of OVERFLOW for Wing-Tip Vortices
,”
AIAA
Paper No. 2016-1782.10.2514/6.2016-1782
6.
Bensow
,
R. R.
, and
van den Boogaard
,
M.
,
2019
, “
Using a PANS Simulation Approach for the Transient Flow Around the Japan Bulk Carrier
,”
J. Ship Res.
,
63
(
2
), pp.
123
129
.10.5957/JOSR.09180064
7.
Alin
,
N.
,
Bensow
,
R. E.
,
Fureby
,
C.
,
Huuva
,
T.
, and
Svennberg
,
U.
,
2010
, “
Current Capabilities of DES and LES for Submarines at Straight Course
,”
J. Ship Res.
,
54
(
3
), pp.
184
196
.10.5957/jsr.2010.54.3.184
8.
Zhang
,
S.
,
Wang
,
L.
,
Yang
,
S.-Z.
, and
Yang
,
H.
,
2010
, “
Numaerical Evaluation of Wind Loads on Semi-Submersible Platform by CFD
,”
ASME
Paper No. OMAE2010-20209.10.1115/OMAE2010-20209
9.
Lefevre
,
C.
,
Constantinides
,
Y.
,
Kim
,
J. W.
,
Gordon
,
R.
,
Jang
,
H.
, and
Wu
,
G.
,
2013
, “
Guidelines for CFD Simulations of Spar VIM
,”
ASME
Paper No. OMAE2013-10333.10.1115/OMAE2013-10333
10.
Liu
,
M.
,
Xiao
,
L.
,
Yang
,
J.
, and
Tian
,
X.
,
2017
, “
Parametric Study on the Vortex-Induced Motions of Semi-Submersibles: Effect of Rounded Ratios of the Column and Pontoon
,”
Phys. Fluids
,
29
(
5
), p.
055101
.10.1063/1.4983347
11.
Pereira
,
F. S.
,
Eça
,
L.
, and
Vaz
,
G.
,
2019
, “
Investigating the Effect of the Closure in Partially-Averaged Navier–Stokes Equations
,”
ASME J. Fluids Eng.
,
141
(
12
), p.
121402
.10.1115/1.4043539
12.
Pereira
,
F. S.
,
Eça
,
L.
,
Vaz
,
G.
, and
Girimaji
,
S. S.
,
2019
, “
On the Simulation of the Flow Around a Circular Cylinder at Re = 140,000
,”
Int. J. Heat Fluid Flow
,
76
, pp.
40
56
.10.1016/j.ijheatfluidflow.2019.01.007
13.
Wang
,
J.
, and
Wan
,
D.
,
2020
, “
Application Progress of Computational Fluid Dynamic Techniques for Complex Viscous Flows in Ship and Ocean Engineering
,”
J. Mar. Sci. Appl.
,
19
(
1
), pp.
1
16
.10.1007/s11804-020-00124-8
14.
Lu
,
N.-X.
,
Bensow
,
R. E.
, and
Bark
,
G.
,
2014
, “
Large Eddy Simulation of Cavitation Development on Highly Skewed Propellers
,”
J. Mar. Sci. Technol.
,
19
(
2
), pp.
197
214
.10.1007/s00773-013-0240-3
15.
Kumar
,
P.
, and
Mahesh
,
K.
,
2017
, “
Large Eddy Simulation of Propeller Wake Instabilities
,”
J. Fluid Mech.
,
814
, pp.
361
396
.10.1017/jfm.2017.20
16.
Guilmineau
,
E.
,
Deng
,
G. B.
,
Leroyer
,
A.
,
Queutey
,
P.
,
Visonneau
,
M.
, and
Wackers
,
J.
,
2018
, “
Numerical Simulations for the Wake Prediction of a Marine Propeller in Straight-Ahead Flow and Oblique Flow
,”
ASME J. Fluids Eng.
,
140
(
2
), p.
021111
.10.1115/1.4037984
17.
Pereira
,
F. S.
,
Eça
,
L.
, and
Vaz
,
G.
,
2019
, “
Simulation of Wingtip Vortex Flows With Reynolds-Averaged Navier–Stokes and Scale-Resolving Simulation Methods
,”
AIAA J.
,
57
(
3
), pp.
932
948
.10.2514/1.J057512
18.
Asnaghi
,
A.
,
Svennberg
,
U.
, and
Bensow
,
R. E.
,
2020
, “
Large Eddy Simulations of Cavitating Tip Vortex Flows
,”
Ocean Eng.
,
195
, p.
106703
.10.1016/j.oceaneng.2019.106703
19.
Liebrand
,
R.
,
Klapwijk
,
M.
,
Lloyd
,
T.
, and
Vaz
,
G.
,
2021
, “
Transition and Turbulence Modeling for the Prediction of Cavitating Tip Vortices
,”
ASME J. Fluids Eng.
,
143
(
1
), p.
011202
.10.1115/1.4048133
20.
Dimonte
,
G.
,
Youngs
,
D. L.
,
Dimits
,
A.
,
Weber
,
S.
,
Marinak
,
M.
,
Wunsch
,
S.
,
Garasi
,
C.
,
2004
, “
A Comparative Study of the Turbulent Rayleigh–Taylor Instability Using High-Resolution Three-Dimensional Numerical Simulations: The Alpha-Group Collaboration
,”
Phys. Fluids
,
16
(
5
), pp.
1668
1693
.10.1063/1.1688328
21.
Thornber
,
B.
,
Drikakis
,
D.
,
Youngs
,
D. L.
, and
Williams
,
R. J. R.
,
2011
, “
Growth of a Richtmyer-Meshkov Turbulent Layer After Reshock
,”
Phys. Fluids
,
23
(
9
), p.
095107
.10.1063/1.3638616
22.
Banerjee
,
A.
, and
Andrews
,
M. J.
,
2009
, “
3D Simulations to Investigate Initial Condition Effects on the Growth of Rayleigh–Taylor Mixing
,”
Int. J. Heat Mass Transfer
,
52
(
17–18
), pp.
3906
3917
.10.1016/j.ijheatmasstransfer.2009.03.032
23.
Pereira
,
F. S.
,
Grinstein
,
F. F.
,
Israel
,
D. M.
,
Rauenzahn
,
R.
, and
Girimaji
,
S. S.
,
2021
, “
Partially Averaged Navier-Stokes Closure Modeling for Variable-Density Turbulent Flow
,”
Phys. Rev. Fluids
,
6
(
8
), p.
084602
.10.1103/PhysRevFluids.6.084602
24.
Pereira
,
F. S.
,
Grinstein
,
F. F.
,
Israel
,
D. M.
, and
Rauenzahn
,
R.
,
2021
, “
Molecular Viscosity and Diffusivity Effects in Transitional and Shock-Driven Mixing Flows
,”
Phys. Rev. E
,
103
(
1
), p.
013106
.10.1103/PhysRevE.103.013106
25.
Grinstein
,
F. F.
,
Saenz
,
J. A.
,
Rauenzahn
,
R. M.
,
Germano
,
M.
, and
Israel
,
D. M.
,
2020
, “
Dynamic Bridging Modeling for Coarse Grained Simulations of Shock Driven Turbulent Mixing
,”
Comput. Fluids
,
199
, p.
104430
.10.1016/j.compfluid.2020.104430
26.
deBruynKops
,
S. M.
, and
Riley
,
J. J.
,
2001
, “
Mixing Models for Large-Eddy Simulation of Nonpremixed Turbulent Combustion
,”
ASME J. Fluids Eng.
,
123
(
2
), pp.
341
346
.10.1115/1.1366679
27.
De
,
A.
, and
Acharya
,
S.
,
2009
, “
Large Eddy Simulation of Premixed Combustion With a Thickened-Flame Approach
,”
ASME J. Eng. Gas Turbines Power
,
131
(
1
), p.
061501
.10.1115/1.3094021
28.
Hasse
,
C.
,
2016
, “
Scale-Resolving Simulations in Engine Combustion Process Design Based on a Systematic Approach for Model Development
,”
Int. J. Engine Res.
,
17
(
1
), pp.
44
62
.10.1177/1468087415597842
29.
Sidhart
,
G. S.
,
Kartha
,
A.
, and
Candler
,
G. V.
,
2016
, “
Filtered Velocity Based LES of Mixing in High Speed Recirculating Shear Flow
,”
AIAA
Paper No. 2016-3184.10.2514/6.2016-3184
30.
Puggelli
,
S.
,
Bertini
,
D.
,
Mazzei
,
L.
, and
Andreini
,
A.
,
2017
, “
Assessment of Scale-Resolved Computational Fluid Dynamics Methods for the Investigation of Lean Burn Spray Flames
,”
ASME J. Eng. Gas Turbines Power
,
139
(
2
), p.
021501
.10.1115/1.4034194
31.
Leudesdorff
,
W.
,
Unger
,
T.
,
Janicka
,
J.
, and
Hasse
,
C.
,
2019
, “
Scale-Resolving Simulations for Combustion Process Development
,”
MTZ Worldwide
,
80
(
2
), pp.
62
67
.10.1007/s38313-018-0136-z
32.
Spalart
,
P. R.
,
Jou
,
W.-H.
,
Strelets
,
M.
, and
Allmaras
,
S. R.
,
1997
, “
Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach
,”
Proceedings of the First Air Force Office of Scientific Research (AFOSR) International Conference on DNS/LES
, Ruston, LO, Aug. 4–8, pp.
137
148
.
33.
Spalart
,
P. R.
,
Deck
,
S.
,
Shur
,
M. L.
,
Squires
,
K. D.
,
Strelets
,
M. K.
, and
Travin
,
A.
,
2006
, “
A New Version of Detached-Eddy Simulation, Resistant to Ambiguous Grid Densities
,”
Theor. Comput. Fluid Dyn.
,
20
(
3
), pp.
181
195
.10.1007/s00162-006-0015-0
34.
Shur
,
M. L.
,
Spalart
,
P. R.
,
Strelets
,
M. K.
, and
Travin
,
A. K.
,
2008
, “
A Hybrid RANS-LES Approach With Delayed-DES and Wall-Modelled LES Capabilities
,”
Int. J. Heat Fluid Flow
,
29
(
6
), pp.
1638
1649
.10.1016/j.ijheatfluidflow.2008.07.001
35.
Kok
,
J. C.
,
Dol
,
H. S.
,
Oskam
,
B.
, and
van der Ven
,
H.
,
2004
, “
Extra-Large Eddy Simulation of Massively Separated Flows
,”
AIAA
Paper No. 2004-264.10.2514/6.2004-264
36.
Menter
,
F.
,
2018
, “
Stress-Blended Eddy Simulation (SBES)—A New Paradigm in Hybrid RANS-LES Modeling
,”
Progress in Hybrid RANS-LES Modelling
. HRLM 2016. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Y. Hoarau, SH. Peng, D. Schwamborn, and A. Revell, eds., Vol. 137. Springer, Cham, Switzerland
.10.1007/978-3-319-70031-1_3
37.
Kok
,
J. C.
,
2017
, “
A Stochastic Backscatter Model for Grey-Area Mitigation in Detached Eddy Simulations
,”
Flow, Turbul. Combust.
,
99
(
1
), pp.
119
150
.10.1007/s10494-017-9809-y
38.
Speziale
,
C. G.
,
1997
, “
Computing Non-Equilibrium Turbulent Flows With Time-Dependent RANS and VLES
,” 15th International Conference on Numerical Methods in Fluid Dynamics (Lecture Notes in Physics, Vol. 490),
P.
Kutler
and
J. F. J.
Chattot
, eds.,
Springer
, Berlin, Germany.
39.
Fasel
,
H. F.
,
Seidel
,
J.
, and
Wernz
,
S.
,
2002
, “
A Methodology for Simulations of Complex Turbulent Flows
,”
ASME J. Fluids Eng.
,
124
(
4
), pp.
933
942
.10.1115/1.1517569
40.
Menter
,
F. R.
,
Kuntz
,
M.
, and
Bender
,
R.
,
2003
, “
A Scale-Adaptive Simulation Model for Turbulent Flow Predictions
,”
AIAA
Paper No. 2003-767.10.2514/6.2003-767
41.
Menter
,
F. R.
, and
Egorov
,
Y.
,
2010
, “
The Scale-Adaptive Simulation Method for Unsteady Turbulent Flow Predictions. Part 1: Theory and Model Description
,”
Flow, Turbul. Combust.
,
85
(
1
), pp.
113
138
.10.1007/s10494-010-9264-5
42.
Girimaji
,
S. S.
,
2006
, “
Partially-Averaged Navier-Stokes Model for Turbulence: A Reynolds-Averaged Navier-Stokes to Direct Numerical Simulation Bridging Method
,”
ASME J. Appl. Mech.
,
73
(
3
), pp.
413
421
.10.1115/1.2151207
43.
Schiestel
,
R.
, and
Dejoan
,
A.
,
2005
, “
Towards a New Partially Integrated Transport Model for Coarse Grid and Unsteady Turbulent Flow Simulations
,”
Theor. Comput. Fluid Dyn.
,
18
(
6
), pp.
443
468
.10.1007/s00162-004-0155-z
44.
Chaouat
,
B.
, and
Schiestel
,
R.
,
2005
, “
A New Partially Integrated Transport Model for Subgrid-Scale Stresses and Dissipation Rate for Turbulent Developing Flows
,”
Phys. Fluids
,
17
(
6
), p.
065106
.10.1063/1.1928607
45.
Smagorinsky
,
J.
,
1963
, “
General Circulation Experiments With the Primitive Equations I. The Basic Experiment
,”
Mon. Weather Rev.
,
91
(
3
), pp.
99
164
.10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
46.
Boris
,
J. P.
,
Grinstein
,
F. F.
,
Oran
,
E. S.
, and
Kolbe
,
R. L.
,
1992
, “
New Insights Into Large Eddy Simulation
,”
Fluid Dyn. Res.
,
10
(
4–6
), pp.
199
228
.10.1016/0169-5983(92)90023-P
47.
Grinstein
,
F. F.
,
Margolin
,
L. G.
, and
Rider
,
W. J.
,
2010
,
Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics
, 2nd ed.,
Cambridge University Press
,
New York
.
48.
The American Society of Mechanical Engineers (ASME),
2009
,
Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer—ASME V&V 20-2009
,
ASME
, New York.
49.
Pereira
,
F. S.
,
Eça
,
L.
,
Vaz
,
G.
, and
Girimaji
,
S. S.
,
2021
, “
Toward Predictive RANS and SRS Computations of Turbulent External Flows of Practical Interest
,”
Arch. Comput. Mech. Eng.
,
28
(
5
), pp.
3953
4029
.10.1007/s11831-021-09563-0
50.
Hamba
,
F.
,
2011
, “
Analysis of Filtered Navier-Stokes Equation for Hybrid RANS/LES Simulation
,”
Phys. Fluids
,
23
(
1
), p.
015108
.10.1063/1.3549933
51.
Roache
,
P. J.
,
1998
,
Verification and Validation in Computational Science and Engineering
, 1st ed.,
Hermosa Publishers
,
Albuquerque, NM
.
52.
The American Society of Mechanical Engineers (ASME), 2016,
Multivariate Metrics—Supplement 1 of ASME V&V 20-2009
,
ASME
, New York (in preparation).
53.
Oberkampf
,
W. L.
, and
Roy
,
C. J.
,
2010
,
Verification and Validation in Scientific Computing
, 1st ed.,
Cambridge University Press
,
Cambridge, UK
.
54.
Pereira
,
F. S.
,
2018
, “
Towards Predictive Scale-Resolving Simulations of Turbulent External Flows
,” Ph.D. thesis, Instituto Superior Técnico, Lisbon, Portugal.
55.
Lawson
,
T. V.
,
Anthony
,
K. C.
,
Cockrell
,
D. J.
,
Davenport
,
A. G.
,
Flint
,
A. R.
,
Freeston
,
D. H.
, and
Harris
,
R. I.
,
1986
, “
Mean Forces, Pressures and Flow Field Velocities for Circular Cylindrical Structures: Single Cylinder With Two-Dimensional Flow
,” Engineering Sciences Data Unit (ESDU), London, Technical Report No. ESDU 80025.
56.
Taylor
,
G. I.
, and
Green
,
A. E.
,
1937
, “
Mechanism of the Production of Small Eddies From Large Ones
,”
Proc. R. Soc. A
,
158
(
895
), pp.
499
521
.10.1098/rspa.1937.0036
57.
Rayleight
,
L.
,
1882
, “
Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density
,”
Proc. London Math. Soc.
,
14
(
1
), pp.
170
177
.10.1112/plms/s1-14.1.170
58.
Taylor
,
G.
,
1950
, “
The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to Their Planes
,”
Proc. R. Soc. A
,
201
(
1065
), pp.
192
196
.10.1098/rspa.1950.0052
59.
Pereira
,
F. S.
,
Grinstein
,
F. F.
,
Israel
,
D. M.
,
Rauenzahn
,
R.
, and
Girimaji
,
S. S.
,
2021
, “
Modeling and Simulation of Transitional Rayleigh-Taylor Flow With Partially Averaged Navier–Stokes Equations
,”
Phys. Fluids
,
33
(
11
), p.
115118
.10.1063/5.0067552
60.
Pereira
,
F. S.
,
Vaz
,
G.
,
Eça
,
L.
, and
Girimaji
,
S. S.
,
2018
, “
Simulation of the Flow Around a Circular Cylinder at Re = 3900 With Partially-Averaged Navier-Stokes Equations
,”
Int. J. Heat Fluid Flow
,
69
, pp.
234
246
.10.1016/j.ijheatfluidflow.2017.11.001
61.
Germano
,
M.
,
1992
, “
Turbulence: The Filtering Approach
,”
J. Fluid Mech.
,
238
, pp.
325
336
.10.1017/S0022112092001733
62.
Pereira
,
F. S.
,
Vaz
,
G.
, and
Eça
,
L.
,
2015
, “
An Assessment of Scale-Resolving Simulation Models for the Flow Around a Circular Cylinder
,”
Proceedings of the Eighth International Symposium on Turbulence, Heat and Mass Transfer (THMT15)
, Sarajevo, Bosnia and Herzegovina, Sept. 15–18, pp.
295
298
.
63.
Silva
,
T. S.
,
Zecchetto
,
M.
, and
da Silva
,
C. B.
,
2018
, “
The Scaling of the Turbulent/Non-Turbulent Interface at High Reynolds Numbers
,”
J. Fluid Mech.
,
843
, pp.
156
179
.10.1017/jfm.2018.143
64.
Pereira
,
F. S.
,
Eça
,
L.
,
Vaz
,
G.
, and
Girimaji
,
S. S.
,
2018
, “
Challenges in Scale-Resolving Simulations of Turbulent Wake Flows With Coherent Structures
,”
J. Comput. Phys.
,
363
, pp.
98
115
.10.1016/j.jcp.2018.02.038
65.
Pereira
,
F. S.
,
Vaz
,
G.
, and
Eça
,
L.
,
2019
, “
Evaluation of RANS and SRS Methods for Simulation of the Flow Around a Circular Cylinder in the Sub-Critical Regime
,”
Ocean Eng.
,
186
, p.
106067
.10.1016/j.oceaneng.2019.05.049
66.
Kamble
,
C.
, and
Girimaji
,
S.
,
2020
, “
Characterization of Coherent Structures in Turbulent Wake of a Sphere Using Partially Averaged Navier–Stokes (PANS) Simulations
,”
Phys. Fluids
,
32
(
10
), p.
105110
.10.1063/5.0024854
67.
Fowler
,
T. S.
,
Witherden
,
F. D.
, and
Girimaji
,
S. S.
,
2020
, “
Partially-Averaged Navier–Stokes Simulations of Turbulent Flow Past a Square Cylinder: Comparative Assessment of Statistics and Coherent Structures at Different Resolutions
,”
Phys. Fluids
,
32
(
12
), p.
125106
.10.1063/5.0027590
68.
Pereira
,
F. S.
,
Grinstein
,
F. F.
,
Israel
,
D. M.
,
Rauenzahn
,
R.
, and
Girimaji
,
S. S.
,
2021
, “
Modeling and Simulation of Transitional Taylor-Green Vortex Flow With Partially Averaged Navier-Stokes Equations
,”
Phys. Rev. Fluids
,
6
(
5
), p.
054611
.10.1103/PhysRevFluids.6.054611
69.
Eça
,
L.
, and
Hoekstra
,
M.
,
2014
, “
A Procedure for the Estimation of the Numerical Uncertainty of CFD Calculations Based on Grid Refinement Studies
,”
J. Comput. Phys.
,
262
, pp.
104
130
.10.1016/j.jcp.2014.01.006
70.
Celik
,
I.
, and
Hu
,
G.
,
2004
, “
Single Grid Error Estimation Using Error Transport Equation
,”
ASME J. Fluids Eng.
,
126
(
5
), pp.
778
790
.10.1115/1.1792254
71.
Drikakis
,
D.
, and
Asproulis
,
N.
,
2011
, “
Quantification of Computational Uncertainty for Molecular and Continuum Methods in Thermo-Fluid Sciences
,”
ASME Appl. Mech. Rev.
,
64
(
4
), p.
040801
.10.1115/1.4006213
72.
Phillips
,
T. S.
, and
Roy
,
C. J.
,
2011
, “
Residual Methods for Discretization Error Estimation
,”
AIAA
Paper No. 2011-3870.10/2514/6.2011-3870
73.
Stern
,
F.
,
Wilson
,
R. V.
,
Coleman
,
H. W.
, and
Paterson
,
E. G.
,
2001
, “
Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures
,”
ASME J. Fluids Eng.
,
123
(
4
), pp.
793
802
.10.1115/1.1412235
74.
Celik
,
I. B.
,
Ghia
,
U.
,
Roache
,
P. J.
,
Freitas
,
C. J.
,
Coleman
,
H. W.
, and
Raad
,
P. E.
,
2008
, “
Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications
,”
ASME J. Fluids Eng.
,
130
(
7
), p.
078001
.10.1115/1.2960953
75.
Rider
,
W.
,
Witkowski
,
W.
,
Kamm
,
J. R.
, and
Wildey
,
T.
,
2016
, “
Robust Verification Analysis
,”
J. Comput. Phys.
,
307
, pp.
146
163
.10.1016/j.jcp.2015.11.054
76.
Xing
,
T.
, and
Stern
,
F.
,
2010
, “
Factors of Safety for Richardson Extrapolation
,”
ASME J. Fluids Eng.
,
132
(
6
), p.
061403
.10.1115/1.4001771
77.
Hills
,
R. G.
,
2006
, “
Model Validation: Model Parameter and Measurement Uncertainty
,”
ASME J. Heat Transfer-Trans. ASME
,
128
(
4
), pp.
339
351
.10.1115/1.2164849
78.
Barmparousis
,
C.
, and
Drikakis
,
D.
,
2017
, “
Multidimensional Quantification of Uncertainty and Application to a Turbulent Mixing Model
,”
Int. J. Numer. Methods Fluids
,
85
(
7
), pp.
385
403
.10.1002/fld.4385
79.
Williamson
,
C. H. K.
,
1996
, “
Vortex Dynamics in the Cylinder Wake
,”
Annu. Rev. Fluid Mech.
,
28
(
1
), pp.
477
539
.10.1146/annurev.fl.28.010196.002401
80.
Zdravkovich
,
M. M.
,
1997
,
Flow Around Circular Cylinders. Volume 1: Fundamentals
, 1st ed.,
Oxford Science Publications
,
Oxford, UK
.
81.
Parnaudeau
,
P.
,
Carlier
,
J.
,
Heitz
,
D.
, and
Lamballais
,
E.
,
2008
, “
Experimental and Numerical Studies of the Flow Over a Circular Cylinder at Reynolds Number 3900
,”
Phys. Fluids
,
20
(
8
), p.
085101
.10.1063/1.2957018
82.
Norberg
,
C.
,
2002
, “
Pressure Distributions Around a Circular Cylinder in Cross-Flow
,” Proceedings of the Symposium on Bluff Body Wakes and Vortex-Induced Vibrations (
BBVIV-3
), Port Arthur, Australia, Dec. 17–20, pp.
1
4
.https://www.ht.energy.lth.se/fileadmin/ht/Norberg-BBVIV3-2002.pdf
83.
Norberg
,
C.
,
2003
, “
Fluctuating Lift on a Circular Cylinder: Review and New Measurements
,”
J. Fluids Struct.
,
17
(
1
), pp.
57
96
.10.1016/S0889-9746(02)00099-3
84.
Brachet
,
M. E.
,
Meiron
,
D. I.
,
Orszag
,
S. A.
,
Nickel
,
B. G.
,
Morf
,
R. H.
, and
Frisch
,
U.
,
1983
, “
Small Scale Structure of the Taylor-Green Vortex
,”
J. Fluid Mech.
,
130
(
1
), pp.
411
452
.10.1017/S0022112083001159
85.
Drikakis
,
D.
,
Fureby
,
C.
,
Grinstein
,
F. F.
, and
Youngs
,
D.
,
2007
, “
Simulation of Transition and Turbulence Decay in the Taylor-Green Vortex
,”
J. Turbul.
,
8
(
20
), p.
N20
.10.1080/14685240701250289
86.
Livescu
,
D.
,
Wei
,
T.
, and
Brady
,
P. T.
,
2021
, “
Rayleigh–Taylor Instability With Gravity Reversal
,”
Phys. D: Nonlinear Phenom.
,
417
, p.
132832
.10.1016/j.physd.2020.132832
87.
MARIN,
2021
, “
ReFRESCO
,” MARIN, Wagegningen, The Netherlands, accessed Oct. 2021, http://www.marin.nl/en/facilities-and-tools/software/refresco
88.
Vaz
,
G.
,
Jaouen
,
F.
, and
Hoekstra
,
M.
,
2009
, “
Free-Surface Viscous Flow Computations. Validation of URANS Code FreSCo
,”
ASME
Paper No. OMAE2009-79398.10.1115/OMAE2009-79398
89.
Pereira
,
F. S.
,
Eça
,
L.
,
Vaz
,
G.
, and
Kerkvliet
,
M.
,
2019
, “
Application of Second-Moment Closure to Statistically Steady Flows of Practical Interest
,”
Ocean Eng.
,
189
, p.
106372
.10.1016/j.oceaneng.2019.106372
90.
Gittings
,
M.
,
Weaver
,
R.
,
Clover
,
M.
,
Betlach
,
T.
,
Byrne
,
N.
,
Coker
,
R.
,
Dendy
,
E.
,
2008
, “
The RAGE Radiation-Hydrodynamic Code
,”
Comput. Sci. Discovery
,
1
(
1
), p.
015005
.10.1088/1749-4699/1/1/015005
91.
Toro
,
E. F.
,
Spruce
,
M.
, and
Speares
,
W.
,
1994
, “
Restoration of the Contact Surface in the HLL-Riemann Solver
,”
Shock Waves
,
4
(
1
), pp.
25
34
.10.1007/BF01414629
92.
Colella
,
P.
, and
Woodward
,
P. R.
,
1984
, “
The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations
,”
J. Comput. Phys.
,
54
(
1
), pp.
174
201
.10.1016/0021-9991(84)90143-8
93.
Thornber
,
B.
,
Mosedale
,
A.
,
Drikakis
,
D.
,
Youngs
,
D.
, and
Williams
,
R. J. R.
,
2008
, “
An Improved Reconstruction Method for Compressible Flows With Low Mach Number Features
,”
J. Comput. Phys.
,
227
(
10
), pp.
4873
4894
.10.1016/j.jcp.2008.01.036
94.
Pereira
,
F. S.
,
2012
, “
Verification of ReFRESCO With the Method of Manufactured Solutions
,” MS thesis, Instituto Superior Técnico, Lisbon, Portugal.
95.
van Leer
,
B.
,
1997
, “
Towards the Ultimate Conservative Difference Scheme
,”
J. Comput. Phys.
,
135
(
2
), pp.
229
248
.10.1006/jcph.1997.5704
96.
Grinstein
,
F. F.
,
Gowardhan
,
A. A.
, and
Wachtor
,
A. J.
,
2011
, “
Simulations of Richtmyer-Meshkov Instabilities in Planar Shock-Tube Experiments
,”
Phys. Fluids
,
23
(
3
), p.
034106
.10.1063/1.3555635
97.
Haines
,
B. M.
,
Grinstein
,
F. F.
, and
Fincke
,
J. R.
,
2014
, “
Three-Dimensional Simulation Strategy to Determine the Effects of Turbulent Mixing on Inertial-Confinement-Fusion Capsule Performance
,”
Phys. Rev. E
,
89
(
5
), p.
053302
.10.1103/PhysRevE.89.053302
98.
Fage
,
A.
, and
Warsap
,
J. H.
,
1930
, “
The Effects of Turbulence and Surface Roughness on the Drag of a Circular Cylinder
,” Aeronautical Research Committee, London, Reports and Memoranda No. 1283.
99.
Sadeh
,
W. Z.
, and
Saharon
,
D. B.
,
1982
, “
Turbulence Effect on Crossflow Around a Circular Cylinder at Subcritical Reynolds Numbers
,” Lewis Research Center, National Aeronautics and Space Administration (NASA), Cleveland, OH, NASA Contractor Report No. 3622.
100.
Norberg
,
C.
, and
Sundén
,
B.
,
1987
, “
Turbulence and Reynolds Number Effects on the Flow and Fluid Forces on a Single Cylinder in Cross Flow
,”
J. Fluids Struct.
,
1
(
3
), pp.
337
357
.10.1016/0889-9746(87)90264-7
101.
Breuer
,
M.
,
2018
, “
Effect of Inflow Turbulence on an Airfoil Flow With Laminar Separation Bubble
,”
Flow, Turbul. Combust.
,
101
(
2
), pp.
433
456
.10.1007/s10494-017-9890-2
102.
West
,
G. S.
, and
Apelt
,
C. J.
,
1982
, “
The Effects of Tunnel Blockage and Aspect Ratio on the Mean Flow Past a Circular Cylinder With Reynolds Numbers Between 104 and 105
,”
J. Fluid Mech.
,
114
(
1
), pp.
361
377
.10.1017/S0022112082000202
103.
Szepessy
,
S.
, and
Bearman
,
P. W.
,
1992
, “
Aspect Ratio and End Plate Effects on Vortex Shedding From a Circular Cylinder
,”
J. Fluid Mech.
,
234
(
1
), pp.
191
217
.10.1017/S0022112092000752
104.
Norberg
,
C.
,
1994
, “
An Experimental Investigation of the Flow Around a Circular Cylinder: Influence of Aspect Ratio
,”
J. Fluid Mech.
,
258
, pp.
287
316
.10.1017/S0022112094003332
105.
Achenbach
,
E.
,
1971
, “
Influence of Surface Roughness on the Cross-Flow Around a Circular Cylinder
,”
J. Fluid Mech.
,
46
(
2
), pp.
321
335
.10.1017/S0022112071000569
106.
Güven
,
O.
,
Farell
,
C.
, and
Patel
,
V. C.
,
1980
, “
Surface-Roughness Effects on the Mean Flow Past Circular Cylinders
,”
J. Fluid Mech.
,
98
(
4
), pp.
673
701
.10.1017/S0022112080000341
107.
Virk
,
D.
,
Hussain
,
F.
, and
Kerr
,
R. M.
,
1995
, “
Compressible Vortex Reconnection
,”
J. Fluid Mech.
,
304
, pp.
47
86
.10.1017/S0022112095004344
108.
Dimonte
,
G.
,
2004
, “
Dependence of Turbulent Rayleigh-Taylor Instability on Initial Perturbations
,”
Phys. Rev. E
,
69
(
5
), p.
056305
.10.1103/PhysRevE.69.056305