A Finite Volume Method Based on a WENO ...

A Finite Volume Method Based on a WENO Reconstruction for Compressible Flows on Hybrid Grids Yidong Xia1, Xiaodong Liu2, and Hong Luo3 North Carolina State University, Raleigh, NC, 27695, USA

A cell-centered finite volume method based on a WENO reconstruction, termed WENO(P0P1) in this paper, is presented for solving the compressible Navier-Stokes equations on 3D hybrid grids. This new WENO(P0P1) method is designed not only to achieve high-order accuracy but also to ensure non-linear stability of the finite volume method. The discretization of the viscous and heat fluxes is carried out using a modified averaging of gradients. The spatially discretized governing equations are integrated in time using a linearized implicit scheme. A fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The parallelization of the WENO(P0P1) method is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model using Message-Passing-Interface (MPI) programming paradigm for distributed memory parallel computing architectures. The developed WENO(P0P1) method is used to compute a variety of flow problems on hybrid grids to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments indicate that this WENO(P0P1) method is able to maintain high accuracy for smooth flows and obtain oscillation-free solutions in the vicinity of discontinuities, and outperforms a van Albada limiter-based finite volume method.

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I. Introduction

omputational Fluid Dynamics has become an indispensable tool for a variety of applications in science and engineering. General speaking, numerical methods used in CFD can be classified by the mesh they use to discretize a computational domain as structured grid methods, unstructured grid methods, and Cartesian grid methods. The structured grid methods alone are not practical for engineering applications, as they have a disadvantage for gridding complex geometries. More often, they are used in the context of Chimera or overlapping approaches1,2 to simplify the grid generation process for a complex configuration. The difficulty of generating a structured grid for complex geometries and the desire in the engineering community to simulate numerically flows past increasingly complex geometries have fueled interest in the development of unstructured grid methods3-7. Unstructured grids provide great flexibility in dealing with the complex geometries encountered in practice and offer a natural framework for solution-adaptive mesh refinement. However, the computational costs and memory requirements for unstructured triangular/tetrahedral grids are generally higher than for structured grids. Although the solution accuracy may not be strongly affected by element type even in the boundary layers, computational efficiency can benefit substantially through the use of prismatic elements in the boundary layers and Cartesian cells in the inviscid regions. This is due to a simple fact that approximately, five to six times more tetrahedra than hexahedra are required to fill a given region with a fixed number of nodes. Although the boundary layers regions occupy only a small portion of the computational domain, it is not uncommon for more than half of the mesh resolution to be packed into this small region, and thus the quadrilateral elements in 2D and prismatic elements in 3D can lead to a significant saving in both memory requirements and computational costs. From the computational efficiency point of view, unstructured triangular/tetrahedral elements should be kept minimal. The advantages of the Cartesian grid approaches8-12 include ease of grid generation, lower computational storage requirements, and significantly less operational count per cell. However, the main challenge in using Cartesian methods is how to deal with arbitrary boundaries, as the grids are not body-aligned. The cells of a Cartesian mesh near the body can extend through surfaces of boundaries. Accurate means of representing boundary conditions in cells that intersect surfaces 1

Post-Doc research associate, Department of Mechanical and Aerospace Engineering, Member AIAA. PhD student, Department of Mechanical and Aerospace Engineering, Student Member AIAA. 3 Professor, Department of Mechanical and Aerospace Engineering, Associate Fellow AIAA. 1 American Institute of Aeronautics and Astronautics 2

are essential for successful Cartesian methods. Since each grid type has its own advantages and disadvantages, the best grid approach is clearly a hybrid one that combines the advantages and strengths of all these three grid types. Finite volume (FV) methods are probably the most successful class of spatial discretization techniques in computational fluid dynamics, due to their relative simplicity and great flexibility. In recent years, significant progress has been made in developing FV methods to solve the compressible Navier-Stokes equations on hybrid grids13-21. Nowadays, the FV methods are widely and routinely used for solving flow problems of scientific and industrial interest22. There exist two major classes of the finite volume methods: cell-centered and vertex-centered finite volume methods. Although the debate that which one has an edge over the other will probably never be settled, a cell-centered finite volume method is preferred in the context of arbitrary grids, which may contain hanging nodes. The existence of several cell types in a hybrid grid poses a great challenge to numerical methods. Although it is unavoidable to treat different cell types differently during the pre- and post-processing stages, it is undesirable that an algorithm depends on a mesh topology during the flow solution stage. The required conditional statements not only lead to an untidy code but also adversely affect speed of programs. Therefore, it is desirable to design an algorithm that treats different cell types in the same way. Such an algorithm is termed grid-transparent in the literature, which does not require any information on the local cell topology. A grid-transparent scheme has a number of advantages: first of all it can significantly reduce the discretization stencils compared to a non-gridtransparent scheme; secondly it can increase the speed of programs; and last it can facilitate the implementation of implicit schemes and parallelization. A first order cell-centered FV method is in general stable on arbitrary grids. However, the second-order finite volume methods based on a piecewise linear reconstruction suffer from the so-called linear instability on unstructured tetrahedral grids, when the reconstruction stencil only involves adjacent face-neighboring cells32. One way to overcome this problem is to use extended stencils, which unfortunately will sacrifice the compactness of the underlying data structure. Furthermore, these linear reconstruction-based FV methods suffer from non-physical oscillations in the vicinity of strong discontinuities for convection-dominant flows. Alternatively, total variation diminishing (TVD) or essentially non-oscillatory (ENO)/weighted ENO(WENO) reconstruction methods can be used to obtain a linear polynomial solution. The TVD and ENO/WENO high order methods are designed to suppress the spurious oscillations in the vicinity of discontinuities, and enhance the order of accuracy of the underlying firstorder finite volume methods, thus achieving both linear and non-linear stability. Slope limiters are widely used in the FV methods to modify the piecewise linear reconstruction and thus to satisfy TVD condition. Unfortunately, the use of limiters will reduce the order of accuracy to first order in the presence of local extrema. Indeed, the limiters to enhance TVD/MUSCL conditions are less robust than the strategies of essential ENO/WENO reconstruction. The ENO schemes were initially introduced by Harten et al.33 in which oscillations up to the order of the truncation error are allowed to overcome the drawbacks and limitations of limiter-based schemes. ENO schemes avoid interpolation across high-gradient regions through biasing of the reconstruction. This biasing is achieved by reconstructing the solution on several stencils at each location, and selecting the reconstruction, which is in some sense the smoothest. This allows ENO schemes to retain higher-order accuracy near high-gradient regions. However, the selection process can lead to convergence problems and loss of accuracy in regions with smooth solution variations. To counter these problems, the so-called weighted ENO scheme introduced by Liu et al.34 is designed to present better convergence rate for steady state problems, better smoothing for the flux vectors, and better accuracy using the same stencils than the ENO scheme. WENO scheme uses a suitably weighted combination of all reconstructions rather than just the one that is judged to be the smoothest. The weighting is designed to favor the smooth reconstruction in the sense that its weight is small, if the oscillation of a reconstructed polynomial is high and its weight is order of one, if a reconstructed polynomial has low oscillation. The development of ENO/WENO-based finite volume methods has been and still remains one of the active research topics in computational fluid dynamics as witnessed by abundance of literatures35-40. The objective of the effort discussed in this paper is to develop an accurate and efficient cell-centered finite volume method based on a simple and new WENO reconstruction: WENO(P0P1), for solving compressible flow problems on arbitrary grids. The WENO(P0P1) method is designed not only to enhance the accuracy of the finite volume method, but also to avoid spurious oscillations in the vicinity of strong discontinuities. In this WENO(P0P1) method, a piecewise linear polynomial solution (P1) is obtained from the underlying piecewise constant finite volume solution (P0) through a WENO reconstruction, which is achieved in two steps: (1) all first derivatives (slopes) on a cell are first obtained from its adjacent face-neighboring cells via 1-exact least-squares reconstruction; (2) the final first derivatives on each cell are then obtained using a WENO strategy based on the reconstructed first 2 American Institute of Aeronautics and Astronautics

derivatives on the cell itself and its adjacent face-neighboring cells. The computation of the viscous fluxes is based on a modified averaging of gradients. The spatially discretized governing equations are integrated in time using a linearized implicit scheme. A fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The parallelization of the present finite volume method is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model using Message-PassingInterface (MPI) programming paradigm for distributed memory parallel computing architectures. The developed WENO(P0P1) method is used to compute a variety of flow problems on hybrid grids to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments indicate that the WENO(P0P1) method is able to maintain the high accuracy for smooth flows and obtain oscillation-free solutions in the vicinity of discontinuities and in general is more accurate and robust than a TVD-based finite volume method. The outline of this paper is as follows. In section 2, the Reynolds-averaged Navier-Stokes equations are described. The numerical method for solving the governing equations is presented in section 3. The computational results for a variety of compressible flow problems and comparison of numerical solutions with available analytical solution and experimental data are presented in section 4. Finally, conclusions are summarized in section 5.

II. Governing Equations The Reynolds-averaged Navier-Stokes equations governing unsteady compressible viscous flows over any region Ω with boundary Γ=∂Ω can be expressed in integral form as

∂ UdΩ + ∫ F j ⋅ n j dΓ = ∫ G j ⋅ n j dΓ ∂t Ω∫ Γ Γ

(2.1)

where the summation convention has been used, and nj is the outward unit normal to Γ. The conservative variable vector U, inviscid flux vector F, and viscous flux vector G, are defined by

⎛ ρ ⎞ ⎜ ⎟ U = ⎜ ρui ⎟ ⎜ ρe ⎟ ⎝ ⎠

⎛ ⎞ ρu j ⎜ ⎟ F j = ⎜ ρu i u j + pδ ij ⎟ ⎜ u ( ρe + p) ⎟ ⎝ j ⎠

⎛ ⎞ 0 ⎜ ⎟ G j = ⎜ σ ij ⎟ ⎜ u σ + q ⎟ j ⎠ ⎝ l lj

(2.2)

Here ρ, p, and e denote the density, pressure, and specific total energy of the fluid, respectively, and ui is the velocity of the flow in the coordinate direction xi. The pressure can be computed from the equation of state

1 p = (γ − 1) ρ (e − u j u j ) 2

(2.3)

which is valid for perfect gas, where γ is the ratio of the specific heats. The components of the viscous stress tensor σij and the heat flux vector are given by

σ ij = µ(

∂u i ∂u j 2 ∂u + ) − µ k δ ij ∂x j ∂xi 3 ∂x k

qj =

1 µ ∂T γ − 1 Pr ∂x j

(2.4)

In the above equations, T is the temperature of the fluid, Pr the laminar Prandtl number, which is taken as 0.7 for air. µ represents the molecular viscosity, which can be determined through Sutherland’s law 3

µ ⎛ T ⎞ 2 T0 + S = ⎜ ⎟ µ 0 ⎜⎝ T0 ⎟⎠ T + S

(2.5)

µ0 denotes the viscosity at the reference temperature T0, and S is a constant which for are assumes the value S = 110oK. The temperature of the fluid T is determined by

T= γ

p ρ

(2.6)

Neglecting viscous effects, the left-hand side of Eq. (2.1) represents the Euler equations governing unsteady compressible inviscid flows. 3 American Institute of Aeronautics and Astronautics

III. Numerical Method The system of the governing Navier-Stokes equations is discretized in space using a cell-centered finite volume method. In a finite volume method, the computational domain Ω is divided by a set of non-overlapping control volumes Ωi, which can be one or combination of the four most common element types: tetrahedra, prisms, pyramids, and hexahedra. On each control volume, the integral form of the governing equations is required to be satisfied,

∂U

∫ ∂t dV + ∫ F n dΓ = ∫ G n dΓ j

Ωi

j

j

∂Ωi

j

(3.1)

∂Ωi

By taking the unknowns to be the cell-averaged conservative variable vector Ui defined by

Ui =

1 Vi

∫ UdV

(3.2)

Ωi

and approximating the boundary integration in Eq. (3.1) using one point quadrature at the midpoint of the face, the semi-discrete form of the equations may be written as

Vi

dU i + ∑ Fij ⋅ n ij Γ ij = ∑ G ij ⋅ n ij Γ ij dt j j

(3.3)

where Vi is the volume of the control volume i, nij is the unit normal vector to the interface ij between the control volume i and its adjacent control volume j, Γij is the area of interface ij (=∂Ωi∩∂Ωj), and Fij and Gij are the inviscid and viscous flux functions at the interface ij , respectively. The inviscid and viscous flux functions are computed differently using upwind and central differencing methods, respectively, which are described in the next two sections. Equation (3.3) represents a system of ordinary differential equations. A fully implicit time scheme is employed to integrate Equation (3.3) to reach steady-state solutions. An approximate Newton method is used to linearize the equations arising from the implicit discretization. A fast, matrix-free implicit method, GMRES+LU-SGS method2324 , is then used to solve the resultant system of linear equations. The parallelization of the present finite volume method is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model using Message-Passing-Interface (MPI) programming paradigm for distributed memory parallel computing architectures. As our design goal is to develop a grid transparent finite volume method on mixed element type grids with possible hanging nodes, the data structure of the flow solver is based on the face of grid, rather than on the elements. A face based data structure has the advantage there are no limitations on the number of faces, which can be connected to an element, and thus can handle hanging nodes with ease. The computation of fluxes and their contribution to cells can be effectively implemented by loops over the faces through gather/scatter operations. 3.1 Discretization of inviscid fluxes If the inviscid fluxes Fij at the interface in equation (3.3) are simply evaluated as an arithmetical average of the fluxes at cell i, and cell j or computed using an arithmetical average of the flow variables at cell i, and cell j, the resulting finite volume method, equivalent to a central difference scheme, allows for the appearance of checkerboarding modes, and thus suffers from linear instability, unless some type of numerical dissipation in the form of artificial viscosity is introduced. To construct a stable scheme and thus ensure linear stability of a finite volume method for the compressible Euler equations, any of the Riemann solvers can be formulated by adopting different forms for the numerical fluxes at the interface. In our work, a number of numerical schemes, including Roe's flux difference splitting25, AUSM family26, HLLC27, and Edwards' low-diffusion flux-splitting scheme28,29 have been implemented to compute the inviscid fluxes at the interfaces. If the cell-averaged variables are used to compute the numerical fluxes, the resulting upwind finite volume method is only first order accurate in space. A higher order of accuracy can be achieved using a reconstruction scheme, which consists of finding a polynomial representation to the solution in each control volume, giving the cell-averaged solutions in each control volume. For the linear reconstruction where a linear polynomial solution on each cell is reconstructed using cell-averaged values of the flow variables in the neighboring cells, the computation of the gradients of the flow variables in the control 4 American Institute of Aeronautics and Astronautics

volume is simply required. Our WENO-based reconstruction is designed not only to increase the accuracy of the finite volume method, but also to avoid spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of a finite volume method. The reconstruction of the solution slopes consists of two steps: a linear polynomial solution is first reconstructed using a least-squares method from the underlying piecewise constant finite volume solution, and the final linear polynomial solution is then obtained using a WENO reconstruction, which is necessary to ensure the nonlinear stability of the second order finite volume method. The resulting finite volume method is referred to as WENO(P0P1) in this paper, where the linear polynomial solution is obtained from the underlying piecewise constant finite volume solution via a WENO reconstruction, which are described in the next two sub-sections. 3.1.1 Least-squares reconstruction One of the most commonly used and simplest reconstruction schemes is the least-squares reconstruction, where the computation of gradients is performed in the form of a minimization problem. The complete details of this reconstruction procedure can be found in Reference 41. However, the procedure is summarized here for completeness. Consider a cell i and assume that the solution varies linearly in the union of the cell i, and its faceneighboring cell j. Then, the change in cell-centered values of the solution can be computed by

∇Ui • (r j − ri ) = U j − Ui

(3.4)

where ri and rj are the position vector for the center of cells i and j, respectively. Similar equations could be written for all adjacent cells that share a face with the cell i subject to an arbitrary weighting factor wi. This yields the following non-square matrix,

⎛ w1 ( x1 − xi ) w1 ( y1 − yi ) w1 ( z1 − zi ) ⎞⎛ U x ⎞ ⎛ w1 (U1 − U i ) ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ! ! ! U = ! ⎜ ⎟⎜ y ⎟ ⎜ ⎟ ⎜ w ( x − x ) w ( y − y ) w ( z − z ) ⎟⎜ U ⎟ ⎜ w (U − U ) ⎟ i n n i n n i ⎠⎝ z ⎠ i ⎠ ⎝ n n ⎝ n n

(3.5)

where n is the number of the face-neighboring cells for the cell i, numbered from 1 to n. Equation 3.5 can be solved using the least-squares method. The algorithm can be implemented using the face-based data structure. This formulation provides a freedom in the choice of weighting coefficients wi. These weighting coefficients can be selected as a function of the geometry and/or solution. Classical approximations in one dimension can be recovered by choosing geometrical weights of the form wi=1/|rj-ri|t for values of t=0,1,2. The numerical experiments in the next section are performed using t=1. For a tetrahedral, the number of face-neighboring cells is four. Consequently, the size of the resulting non-square matrix is 4x3. In the present work, this over-determined linear system of 4 equations for 3 unknowns is solved in the least-squares sense using either normal equation approach or the QR decomposition to obtain the first derivatives of the reconstructed linear polynomial solution. One can easily verify that this least-squares reconstruction satisfies the so-called 1-exactness, i.e., it can reconstruct a linear polynomial function exactly. 3.1.2 WENO reconstruction This least-squares reconstructed finite volume method can be successfully used to solve the 2D compressible Euler equations for smooth flows on arbitrary grids and is able to achieve the designed second order of accuracy and significantly improve the accuracy of the underlying first-order FV method. However, when extended to solve the 3D compressible Euler equations on tetrahedral grids, this cell-centered finite volume method suffers from the socalled linear instability, which occurs even for the linear hyperbolic equation33. This linear instability is attributed to the fact that the reconstruction stencils only involve von Neumann neighborhood, i.e., adjacent face-neighboring cells33. The linear stability can be achieved using extended stencils, which will unfortunately sacrifice the compactness of the underlying FV methods. Furthermore, such a linear reconstruction-based FV method cannot maintain the non-linear instability, leading to non-physical oscillations in the vicinity of strong discontinuities. Alternatively, ENO/WENO can be used to reconstruct a linear polynomial solution, which can not only enhance the order of accuracy of the underlying FV method but also achieve both linear and non-linear stability. Specifically, the WENO scheme introduced by Dumber et al.36,37 is adopted in this work, where an entire linear polynomial solution 5 American Institute of Aeronautics and Astronautics

on cell i is obtained using a nonlinear WENO reconstruction as a convex combination of the least-squares reconstructed first derivatives at the cell itself (k=0) and its face-neighboring cells (k=1,..,nface),

∂U ∂xn

nface

WENO i

=

∑w

k

k =0

∂U ∂xn

k

(3.6)

where nface is the number of the face-neighboring cells for cell i and the normalized nonlinear weights wk are computed as

wk =

~ w k

nface

∑ w~

(3.7)

k

The non-normalized nonlinear weights indicator oi

i =0 ~ are functions of the linear weights λi and the so-called oscillation w i

~ = w i

λi (ε + oi )γ

(3.8)

where ε is a small positive number used to avoid division by zero, and γ an integer parameter to control how fast the non-linear weights decay for non-smooth stencils. The oscillation indicator for the reconstructed second order polynomials is simply defined as

ok = [(

∂U ∂xn

k

) 2 ]1 / 2

(3.9)

The linear weights λi can be chosen to balance the accuracy and the non-oscillatory property of the FV method. Note that the least-squares reconstructed polynomial at the cell itself serves as the central stencil and the least-squares reconstructed polynomials on its face-neighboring cells act as biased stencils in this WENO reconstruction. This reconstructed linear polynomial solution is then used to compute the fluxes at the interfaces in Eq. (3.3). As demonstrated in the next section, the resulting WENO(P0P1) method is able to significantly improve the accuracy of the underlying first-order FV method, and suppress spurious oscillations in the vicinity of discontinuities, and maintain both linear and non-linear stability. Note that the WENO(P0P1) method is not compact anymore, as neighbor's neighbors are used in the solution update. However, the stencil used in the reconstruction is compact, involving only von Neumann neighbors. Consequently, the WENO(P0P1) method can be implemented in a compact manner 3.2 Discretization of viscous fluxes The finite volume approximation of the viscous fluxes requires the evaluation of the both flow variables u and their derivatives at the interface. The flow variables at the interfaces can be computed using the following three approaches: 1) take the simple arithmetic average of the cell-averaged variables at the two neighboring cells; 2) take the simple arithmetic average of the reconstructed and limited variables at the two neighboring cells, and 3) take the values from interpolation of the variables at the two neighboring cells. Unfortunately, the first approach is clearly dangerous and inconsistent from the analysis in Reference 42, although this is a common practice to compute flow variables at the interface for the viscous fluxes. The second approach might introduce local extrema, causing instability. The third approach is not second order accurate, as long as the interface is not orthogonal to the position vector connecting the two neighboring cells. From the analysis given in Reference 42, it is natural to evaluate flow variables in the viscous fluxes at the interfaces using the inverse-distance weighting of the variables from the two neighboring cells. There are two popular approaches in the literature to estimate the gradients of the flow quantities at the cell interfaces: gradient computation based on divergence theorem using so-called diamond control volume and gradient computation based on a modified averaging of gradients. The former approach is unfortunately not grid-transparent and will not be considered in this work. For the latter, an initial value for the gradient at the cell interfaces may be obtained by averaging the values of two adjacent cells, φi and φj, to give

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1 ( ∇φ)ij = [( ∇φ)i + ( ∇φ) j ] 2

(3.10)

Subsequently a correction in the direction Sij is applied to avoid odd-even decoupling, where

! ! rj − ri ! sij = ! ! |rj − ri |

(3.11)

This leads to the following gradient on the interface

⎡ ! φ j − φi ⎤ ! ( ∇φ)ij = ( ∇φ)ij − ⎢( ∇φ)ij ⋅ sij − ! ! ⎥ ⋅ sij |rj − ri | ⎥⎦ ⎢⎣

(3.12)

This modification results in a stable discretization of the viscous terms, and does not affect the accuracy as long as the interface integration point is located halfway between the two cell-centers. Unfortunately, this is not the case on a non-uniform grid, rendering this type of viscous discretization highly mesh dependent. Similar to evaluate flow quantities at the cell interfaces, an inverse distance weighting is used to evaluate the value of the gradient at the interfaces instead of using a simple arithmetic averaging. Although this method is not compact, which involves the neighbors of the neighbors due to the averaging of gradients, it is grid transparent. Deister has shown that this method reveals a smaller inconsistency error than the other schemes and remains positive, even for locally refined grids. This type of the viscous flux discretization is clearly grid-transparent. It should be pointed out that a second order accurate, consistent, and grid-transparent, cell-centered finite volume method for the discretization of the viscous fluxes on a non-uniform and highly distorted grid remains at large, to the best of our knowledge. IV. Numerical Examples Computational results for a few test cases are presented in this section. Wherever possible, the solutions are also presented using a second order finite volume method with van Albada limiter, termed Albada(P0P1) hereinafter, in an effort to demonstrate the performance of the WENO(P0P1) method. Since the present WENO(P0P1) method is implemented on arbitrary grids, our finite volume code has the ability to compute 1D, 2D, and 3D problems, which greatly alleviates the need and pain for code maintenance and upgrade. Results for one-dimensional flows can be readily obtained by setting the number of cells in both y- and z-directions to be 1 using a hexahedral grid. For twodimensional problems, the number of cells in the z-direction is simply set to be 1. A. Sod shock tube problem

Figure 1. Computed density contours in the flow field for Sod shock tube problem at t=20. The classic Sod shock tube problem is computed in this test case using 50 cells, which is chosen to validate and verify the implementation of the developed WENO(P0P1) method. The initial conditions for the computation are the following: ρ=1, u=0, p=1 for 0≤x≤50, and ρ=0.125, u=0, p=0.1 for 50≤x≤100. Figure 1 shows the computed density contours in the flow field at t=20. Figure 2 compares the density profile at t=20 between the exact solution and 7 American Institute of Aeronautics and Astronautics

computed solutions using the WENO reconstruction and van Albada limiter, respectively. Both WENO(P0P1) and Albada(P0P1) methods produce very similar results for this test case.

Figure 2. Comparison of computed density profiles obtained by the WENO reconstruction and van Albada limiter with the analytical solution at t=20. B. Subsonic Flow past a Circular Cylinder The problem under consideration is an inviscid subsonic flow past a circular cylinder at a Mach number of M∞=0.38. This test case is chosen to assess the order of accuracy and discretization error of the finite volume method using WENO reconstruction and van Albada limiter. Fig. 3 shows four successively refined o-type hybrid grids having 16x5, 32x9, 64x17, and 128x33 points, respectively. The first number refers to the number of points in the circular direction, and the second designates the number of concentric circles in the mesh. The radius of the cylinder is r1=0.5, the domain is bounded by r33=20, and the radii of concentric circles for 128x33 mesh are set up as

where α=1.1580372. The coarser grids are generated by successively coarsening the finest mesh. Numerical solutions to this problem are computed on these four grids to obtain quantitative measurement of the order of accuracy and discretization errors of the finite volume method, when different weightings are used in the GreenGauss reconstruction. The computed Mach number contours using the WENO reconstruction is shown in Figure 4. In this case, the entropy production is served as the error measurement. Note that the entropy production serves as a good criterion to measure accuracy of the numerical solutions, since the flow under consideration is isentropic. Fig.~5 provides the details of the spatial accuracy of the WENO(P0P1) and Albada(P0P1) methods for this numerical experiment. One can observe that the results obtained using the WENO(P0P1) method appear to be better than those obtained using the Albada(P0P1) method in terms of the order of convergence rate and the absolute error. Unfortunately, none of them exhibits a full second order of convergence even for this simple problem on a welldesigned set of grids and the baseline finite volume method also fails to offer a first order of accuracy, although all of them were able to achieve a consistent convergence.

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Figure 3. Sequences of four successively globally refined meshes 16x5, 32x9, 64x17, 128x33 for computing inviscid subsonic flow past a circular cylinder.

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Figure 4. Computed Mach number contours on four successively refined grids obtained using the WENO reconstruction for subsonic flow past a circular cylinder at a Mach number of 0.38.

Figure 5. Accuracy summary for an inviscid subsonic flow past a circular cylinder obtained by the first order, WENO-based, and van Albada limiter-based second-order finite volume methods. 10 American Institute of Aeronautics and Astronautics

C. Transonic flow past a M6 wing A transonic flow over the ONERA M6 wing geometry is considered in this test case. The M6 wing has a leading edge sweep angle of 30 degree, an aspect of 3.8, and a taper ratio of 0.562. The airfoil section of the wing is the ONERA "D" airfoil, which is a 10% maximum thickness-to-chord ratio conventional section. The flow solutions are presented at a Mach number of 0.84 and an angle of attack of 3.06o using both WENO(P0P1) and Albada(P0P1) methods. The mesh contains 136,705 elements, 25,616 points, and 5,017 boundary faces, shown in Figure 6. Figure 7 shows the computed pressure contours on the upper wing surface obtained by the WENO(P0P1) and Albada(P0P1) methods. The computed pressure coefficient distributions obtained by these two methods are compared at six spanwise stations in Figure 8, where experimental data for the pressure coefficients is also given as a reference. The entropy production distribution on the surface of the airfoil obtained by these two methods is compared in Fig. 9. The numerical solution obtained by the WENO(P0P1) method is more accurate than the one obtained by the Albada(P0P1) method by judging the entropy production distribution on the surface of the wing and the ability to capture the suction peak at the leading edge. Note that the entropy production is directly related to the error of the numerical methods, as it should be zero everywhere before shock waves and for subsonic flows.

Figure 6. Mesh used for computing transonic flow past a M6 wing (nelem=334,898, npoin=61,659, nbfac=20,098)

Figure 7. Computed pressure contours for a transonic flow past a M6wing at a Mach number of 0.84, and an angle of attack of 3.06o obtained by the WENO reconstruction (left) and van Albada limiter (right). 11 American Institute of Aeronautics and Astronautics

Figure 8. Comparison of the computed pressure coefficient distributions at different semispan locations obtained by the WENO(P0P1) and Albada(P0P1) methods with experimental data.

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Figure 9. Comparison of the computed entropy production distributions at different semispan locations obtained by the WENO(P0P1) and Albada(P0P1) methods.

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D. Blasius Boundary Layer The laminar boundary layer over a unit flat plate is considered in this test case. This simple problem is chosen to demonstrate the consistency of the WENO(P0P1) method for solving the Navier-Stokes equations. Two meshes used in this test case have the same number of grid points (121x31), the same distribution of the grid points in the xdirection, but a different distribution of grid points in the y-direction. In order to cluster points near the wall, the point distribution in the y-direction follows a geometric stretching. The stretching ratio is the ratio of the heights of the two successive elements. A stretching ratio of 1.2 and 1.3 is used for the two meshes in the computation, respectively. For the grid with a stretching ratio of 1.2, the height of the first element is 0.548661E-03. When a stretching ratio is set to 1.3, the first grid spacing off the wall is 0.742787E-04. Figure 10 shows one of the grids used in this test case. Figure 11 compares the computed skin friction coefficients obtained by the WENO(P0P1) method on these two grids. As expected, the larger the stretching ratio, the more grid points and the better grid resolution in the boundary layer, and thus the better the numerical solution, clearly demonstrating a consistent, accurate, and convergent solution of our WENO(P0P1) method.

Figure 10. Mesh used for computing the Blasius flat plate boundary layer flow.

Figure 11. Skin friction coefficient obtained by the WENO(P0P1) method for the flat plate boundary layer. 14 American Institute of Aeronautics and Astronautics

E. Laminar Low Reynolds Number Flow Past a Sphere This example, largely studied in literature, is chosen to demonstrates the robustness of the WENO(P0P1) method for low Reynolds number flows. The test case presents a sphere within a laminar flow at Mach number of 0.3 and Reynolds number of 117. At these conditions, a steady state solution should be formed with a large toroidal vortex formed just aft of the sphere. The computational mesh, shown in Figure 12, consists of 200,416 tetrahedral elements, 37,274 points, and 6,128 boundary points. Figure 13 shows the computed Mach number contours around the sphere along with streamlines identifying the toroidal vortex. The measured flow separation occurs at roughly 133o from the stagnation point, which qualitatively agrees with experimental observation.

Figure 12. Mesh used for computing low Reynolds number flow past a sphere. (nelem=334,898, npoin=61,659, nbfac=20,098)

Figure 13. Mach number contours and streamlines for laminar flow past a sphere at a Mach number of 0.3 and Reynolds number of 117.

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F. Implicit Large Eddy Simulation of a Lid-driven Cavity In the final example, a numerical experiment for a lid-driven cavity of square cross-section at a Reynolds number of 10,000 is conducted using the implicit large eddy simulation (LES), where the Spanwise Aspect Ratio (SAR) of the cavity is equal to 0.5:1. The lid-driven cavity presents complex physical phenomena, though the geometry is simple. Therefore it is an adequate example to assess the performance of the developed WENO(P0P1) reconstruction-based finite volume method. For the description of the physics and the reference data among the abundant literature, we refer to the classical experiment conducted by Prasad and Koseff (1989)43, and the LES by Zang et al (1993)44. In this experiment, the cavity dimensions are 1 unit in the streamwise x- and vertical y- directions, and 0.5 unit in the spanwise z- direction. A hexahedral grid consisting of 64×64×32 grid points is used in the computation. The grid distribution is non-uniform in the x- and y-directions, as the grid points are clustered near the walls, but is uniform in the z- direction, as shown in Fig. 14 (a). The grid spacing is geometrically stretched away from the walls with the minimum value hmin equal to 0.005. On the bottom and sidewalls, the no-slip and adiabatic boundary conditions are prescribed. Along the top “lid”, the no-slip and adiabatic boundary conditions along with a lid velocity uB equal to a Mach number of Ma = 0.2 are prescribed. The reference kinematic viscosity ν is calculated based on the prescribed Reynolds number, the lid velocity uB, and the lid length B. The computation is initialized with a zero-velocity field. The ensemble-averaged statistics are collected through running the solver for 1,000 time units with 100 time planes using the explicit three-stage TVD Runge-Kutta time stepping scheme. A fixed time step size of Δt = 0.0005 is used. Fig. 14 (b) shows a snapshot of the instantaneous Mach number iso-surfaces. Figs. 15 (a) – (c) show a series of the contours for the time-averaged variables plotted on the spanwise mid-plane. The distribution of the mean velocity components is plotted along the centerlines on the spanwise mid-plane, as shown in Fig 15 (d). One can see that the WENO(P0P1) based finite volume solution matches closely with most part of the reference experimental and LES data, but is under-resolved near the bottom wall and left sidewall regions, partially due to the coarse mesh used in this computation. Note that this is implicit LES of an incompressible flow problem by using a compressible finite volume method, in which no explicit sub-grid scale model is used.

(a)

(b)

Figure 14. (a) Mesh used for conducting large eddy simulation of a lid-driven cavity at a Reynolds number 10,000 (nelem=123,039, npoin=131,072, nbfac=11,782); (b) Snapshot of the computed instantaneous Mach number iso-surface.

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(a)

(b) 0 1

0.2

x-coordinate 0.4 0.6

0.8

1 1

Exp. (1989) LES (1993) WENO

0.8

/uB

0.6 0 0.4

y-coordinate

0.5

-0.5 0.2

-1 -1

-0.5

(c)

0 /uB

0.5

1

0

(d)

Figure 15. (a) Mean streamtraces in the spanwise mid-plane; (b) mean x-velocity contours in the spanwise mid-plane; (c) mean y-velocity contours in the spanwise mid-plane; (d) distribution of velocity components along the centerlines in the spanwise mid-plane.

V. Conclusions and Outlook A finite volume method based on a new WENO reconstruction has been developed for solving the compressible Navier-Stokes equations on 3D hybrid grids. The WENO reconstruction is designed to achieve high-order accuracy and suppress oscillations in the vicinities of discontinuities, and thus ensure the non-linear stability of the finite volume method. The developed WENO(P0P1) method is used to compute a variety of flow problems on hybrid grids to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments demonstrate that this WENO-based finite volume method is more accurate than the van Albada limiter-based finite volume method. Future work will extend this WENO-based finite volume method to turbulent and chemically reactive flows.

Acknowledgments This research was partially sponsored by the DOES Office of Nuclear Energy’s Nuclear Energy University Program. 17 American Institute of Aeronautics and Astronautics

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