Implementation of some higher-order convection schemes on non-uniform grids

A generalized formulation is applied to implement the quadratic upstream interpolation (QUICK) scheme, the second-order upwind (SOU) scheme and the second-order hybrid scheme (SHYBRID) on non-uniform grids. The implementation method is simple. The accuracy and efficiency of these higher-order schemes on non-uniform grids are assessed. Three well-known bench mark convection-diffusion problems and a fluid flow problem are revisited using non-uniform grids. These are: (1) transport of a scalar tracer by a uniform velocity field; (2) heat transport in a recirculating flow; (3) two-dimensional non-linear Burgers equations; and (4) a two-dimensional incompressible Navier-Stokes flow which is similar to the classical lid-driven cavity flow. The known exact solutions of the last three problems make it possible to thoroughly evaluate accuracies of various uniform and non-uniform grids. Higher accuracy is obtained for fewer grid points on non-uniform grids. The order of accuracy of the examined schemes is maintained for some tested problems if the distribution of non-uniform grid points is properly chosen.     

Treatment of numerical diffusion in strong convective flows

A three-dimensional extension of the QUICK scheme adapted for the finite volume method and non-uniform grids is presented to handle convection-diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three-dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two-dimensional, steady state and the other three-dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.     

A bounded convection scheme for the overlapping control volume approach

This paper introduces a flux-limited scheme FLOCV for the overlapping control volume (OCV) approach to 2D steady and unsteady convection–diffusion problems on structured non-orthogonal grids. FLOCV switches from second- to first-order interpolation in the presence of extrema. Smooth switching between the two is ensured by weighted average second- and first-order upwind differencing, with the weights being dynamically determined. Five convective test problems are solved using this scheme and results are compared with known analytical solutions. It is found that FLOCV approximately retains second-order accuracy of the base discretization scheme on uniform grids and smooth non-uniform orthogonal grids. It is also found effective in removing oscillations for problems with discontinuities on both orthogonal and non-orthogonal grids, with little degradation of accuracy. © 1997 John Wiley & Sons, Ltd.     

Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods

In the framework of a cell-centered finite volume method (FVM), the advection scheme plays the most important role in developing FVMs to solve complicated fluid flow problems for a wide range of Reynolds numbers. Advection schemes have been widely developed for FVMs employing pressure-velocity coupling methodology in the incompressible flow limit. In this regard, the physical influence upwind scheme (PIS) is developed for a cell-centered finite volume coupled solver (FVCS) using a pressure-weighted interpolation method for linking the pressure and velocity fields. The well-known exponential differencing scheme and skew upwind differencing scheme are also deployed in the current FVCS and their numerical results are presented. The accuracy and convergence of the present PIS are evaluated solving flow in a lid-driven square cavity, a lid-driven skewed cavity, and over a backward-facing step (BFS). The flow within the lid-driven square cavity is numerically solved at Reynolds numbers from 400 to 10 000 on a relatively coarse mesh with respect to other reported solutions. The lid-driven skewed cavity test case at Reynolds number of 1000 demonstrates the numerical performance of the present PIS on nonorthogonal grids. The flow over a BFS at Reynolds number of 800 is numerically solved to examine capabilities of current FVCS employing the current PIS in inlet-outlet flow computations. The numerical results obtained by the current PIS are in excellent agreement with those of benchmark solutions of corresponding test cases. Incorporating implicit role of pressure terms in a pressure-weighted interpolation method and development of PIS provides satisfactory solution convergence alongside the numerical accuracy for the current FVCS. A particular numerical verification is performed for the V velocity calculation within the BFS flow field, which confirms the reliability of present PIS.     

Perturbation finite volume method for convective-diffusion integral equation

A perturbation finite volume (PFV) method for the convective-diffusion integral equation is developed in this paper. The PFV scheme is an upwind and mixed scheme using any higher-order interpolation and second-order integration approximations, with the least nodes similar to the standard three-point schemes, that is, the number of the nodes needed is equal to unity plus the face-number of the control volume. For instance, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D linear and nonlinear problems, 2-D and 3-D flow model equations. Comparing with other standard three-point schemes, the PFV scheme has much smaller numerical diffusion than the first-order upwind scheme (UDS). Its numerical accuracies are also higher than the second-order central scheme (CDS), the power-law scheme (PLS) and QUICK scheme. The project supported by the National Natural Science Foundation of China (10272106, 10372106)     

New numerical schemes based on a criterion for constructing essentially stable and accurate numerical schemes for convection-dominated equations

In order to obtain stable and accurate numerical solutions for the convection-dominated steady transport equations, we propose a criterion for constructing numerical schemes for the convection term that the roots of the characteristic equation of the resulting difference equation have poles. By imposing this criterion on the difference coefficients of the convection term, we construct two numerical schemes for the convection-dominated equations. One is based on polynomial differencing and the other on locally exact differencing. The former scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $, which is the critical value for its stability, while it approaches the second-order upwind scheme as Rm goes to infinity. Hence the former scheme interpolates a stable scheme between the QUICK scheme at Rm = $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $ and the second-order upwind scheme at Rm = ∞. Numerical solutions with the present new schemes for the one-dimensional, linear, steady convection-diffusion equations showed good results.     

Numerical aspects of calculation of confined swirling flows with internal recirculation

Predictions were performed for two different confined swirling flows with internal recirculation zones. The convection terms in the elliptic governing equations were discretized using three different finite differencing schemes: hybrid, quadratic upwind interpolation and skew upwind differencing. For each flow case, calculations were carried out with these schemes and successively refined grids were employed. For the turbulent flow case the k-ε turbulence model was used. The predicted cases were a laminar swirling flow investigated by Bornstein and Escudier, and a turbulent low-swirl case studied by Roback and Johnson. In both cases an internal recirculation zone was present. The laminar case is well predicted when account is taken of the estimated radial velocity component at the chosen inlet plane. The quadratic upwind interpolation and skew upwind schemes predict the main features of the internal recirculation zone also with a coarse grid. The turbulent case is well predicted with the coarse as well as the finer grids, the skew upwind and quadratic upwind interpolation schemes yielding results very close to the measurements. It is concluded that the skew upwind scheme reaches grid independence slightly before the quadratic upwind scheme, both considerably earlier than the hybrid scheme.     

Comparison of SIMPLE- and PISO-type algorithms for transient flows

Various pressure-based schemes are proposed for transient flows based on well-established SIMPLE and PISO algorithms. The schemes are applied to the solution of unsteady laminar flow around a square cylinder and steady laminar flow over a backward-facing step. The implicit treatment and the performance of the various schemes are evaluated by using benchmark solutions with a small time step. Three different second-order-accurate time derivatives based on different time levels are presented. The different time derivatives are applied to the various schemes under consideration. Overall the PISO scheme was found to predict accurate results and was robust. However, for small time step values, alternative schemes can predict accurate results for approximately half the computational cost. The choice of time derivative proved to be very significant in terms of the accuracy and robustness of a scheme. Significantly, the one-sided forward differencing scheme was the most successful used in conjunction with a strongly implicit-based algorithm. However, a greater degree of accuracy was achieved using the standard PISO algorithm with the Crank–Nicolson time derivative. Recommendations for future work are discussed. © 1998 John Wiley & Sons, Ltd.     

Development and assessment of a variable-order non-oscillatory scheme for convection term discretization

A new scheme for convection term discretization is developed, called VONOS (variable-order non-oscillatory scheme). The development of the scheme is based on the behaviour of well-known non-oscillatory schemes in the pure convection of a step profile test case. The new scheme is a combination of the QUICK and BSOU (bounded second-order upwind) schemes. These two schemes do not have the same formal order of accuracy and for that reason the formal order of accuracy of the new scheme is variable. The scheme is conservative, bounded and accurate. The performance of the new scheme was assessed in three test cases. The results showed that it is more accurate than currently used higher-order schemes, so it can be used in a general purpose algorithm in order to save computational time for the same level of accuracy. © 1998 John Wiley & Sons, Ltd.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

Calculation of laminar flows with second-order schemes and collocated variable arrangement

A numerical study of laminar flows is carried out to examine the performance of two second-order discretization schemes: a total variation diminishing scheme and a second-order upwind scheme. The former has the same form as the standard first-order hybrid central upwind scheme, but with a numerical diffusion reduced by the Van Leer limiter; the latter is based on the linear extrapolation of cell face values using the two upwind neighbors. A collocated grid arrangement is used; oscillations which could be generated by pressure–velocity decoupling are avoided via the Rhie–Chow interpolation. Two iterative solution methods are used: (i) the deferred correction procedure proposed by Khosla and Rubin and (ii) implicit treatment of the second-order upwind contribution. Three two-dimensional laminar test cases are considered for assessment: the plane lid-driven cavity, the plane backward facing step and the axisymmetric pipe with sudden contraction. Experimental data are available for the two last cases. Both the total variation diminishing and the second-order upwind schemes give wiggle-free results and can predict the flowfields more accurately than the standard first-order hybrid central upwind scheme. © 1998 John Wiley & Sons, Ltd.     

Higher-order Upwind Distribution-formula Scheme for Structured and Unstructured Adaptive Flow Solvers

The development of inviscid and viscous flow solvers for both structured and unstructured meshes is presented in this paper. The solution method is the distribution-formula scheme. This is an explicit, cell-vertex, finite volume method which is essentially second-order accurate in both space and time. The Euler and Navier-Stokes equations are integrated over each finite volume cell to determine the change in flow properties (e.g. density) for the cell. Distribution formulas are then used to distribute such changes to the surrounding vertices. Increments in each vertex (which is a calculation point) thus consist of contributions from the surrounding cells. The original discretization technique involves central differencing and is simple, robust and computationally efficient. In this work, starting with inviscid flow simulations using the original scheme on structured grids, improvements are subsequently made to the scheme by replacing the central differencing portion with MUSCL type higher-order upwind differencing. Numerical investigations with the improved scheme are performed using inviscid flow simulations on structured grids. Upon establishing improved accuracy, stability and excellent shock capturing properties, further extension to viscous flow computations on unstructured adaptive meshes is implemented. Results are presented for laminar, viscous flow over a NACA 0012 airfoil.     

Evaluation of various high-order-accuracy schemes with and without flux limiters

Conventional high-order schemes with reduced levels of numerical diffusion produce results with spurious oscillations in areas where steep velocity gradients exist. To prevent the development of non-physical oscillations in the solution, several monotonic schemes have been proposed. In this work, three monotonic schemes, namely Van Leer's scheme, Roe's flux limiter and the third-order SHARP scheme, are compared and evaluated against schemes without flux limiters. The latter schemes include the standard first-order upwind scheme, the second-order upwind scheme and the QUICK scheme. All the above schemes are applied to four two-dimensional problems: (i) rotation of a scalar ‘cone’ field, (ii) transport of a scalar ‘square’ field, (iii) mixing of a cold with a hot front and (iv) deformation of a scalar ‘cone’ field. These problems test the ability of the selected schemes to produce oscillation-free and accurate results in critical convective situations. The evaluation of the schemes is based on several aspects, such as accuracy, economy and complexity. The tests performed in this work reveal the merits and demerits of each scheme. It is concluded that high-order schemes with flux limiters can significantly improve the accuracy of the results.     

Higher‐order bounded differencing schemes for compressible and incompressible flows

In recent years, three higher‐order (HO) bounded differencing schemes, namely AVLSMART, CUBISTA and HOAB that were derived by adopting the normalized variable formulation (NVF), have been proposed. In this paper, a comparative study is performed on these schemes to assess their numerical accuracy, computational cost as well as iterative convergence property. All the schemes are formulated on the basis of a new dual‐formulation in order to facilitate their implementations on unstructured meshes. Based on the proposed dual‐formulation, the net effective blending factor (NEBF) of a high‐resolution (HR) scheme can now be measured and its relevance on the accuracy and computational cost of a HR scheme is revealed on three test problems: (1) advection of a scalar step‐profile; (2) 2D transonic flow past a circular arc bump; and (3) 3D lid‐driven incompressible cavity flow. Both density‐based and pressure‐based methods are used for the computations of compressible and incompressible flow, respectively. Computed results show that all the schemes produce solutions which are nearly as accurate as the third‐order QUICK scheme; however, without the unphysical oscillations which are commonly inherited from the HO linear differencing scheme. Generally, it is shown that at higher value of NEBF, a HR scheme can attain better accuracy at the expense of computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

多维迎风方法在非结构/混合网格热流计算中的应用研究

非结构/混合网格具有极强的几何灵活性,在复杂外形飞行器的气动力特性数值模拟中已得到广泛应用,但目前还难以准确地预测气动热环境。本文从非结构/混合网格热流计算的三个需求出发,选取了多维迎风方法,并与其他方法进行了对比研究。以二维圆柱高超声速绕流这一Benchmark典型问题为例,对比研究了多维迎风方法和几种广泛使用的无粘通量格式(Roe格式、Van Leer格式和AUSMDV格式)对混合网格热流计算精度的影响。结果表明,多维迎风方法在热流计算精度、鲁棒性以及收敛性方面表现良好。最后,将多维迎风方法应用于常规混合网格上的圆柱和钝双锥绕流问题,均得到了较好的热流计算结果,为非结构/混合网格热流计算在复杂高超飞行器中的应用奠定了基础。     

HIGH-ORDER-ACCURATE SCHEMES FOR INCOMPRESSIBLE VISCOUS FLOW

We present new finite difference schemes for the incompressible Navier–Stokes equations. The schemes are based on two spatial differencing methods; one is fourth-order-accurate and the other is sixth-order accurate. The temporal differencing is based on backward differencing formulae. The schemes use non-staggered grids and satisfy regularity estimates, guaranteeing smoothness of the solutions. The schemes are computationally efficient. Computational results demonstrating the accuracy are presented. © 1997 by John Wiley & Sons, Ltd.     

A robust low‐dissipation AUSM‐family scheme for numerical shock stability on unstructured grids

The simple low‐dissipation advection upwind splitting method (SLAU) scheme is a parameter‐free, low‐dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional‐shock‐detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

Large eddy simulation of flow around a rectangular cylinder

A large eddy simulation (LES) of turbulent flow around a stationary rectangular cylinder at high Reynolds number of 2.2×10⁴ is conducted as the first step to prove the applicability of LES to practical engineering problems. Time-averaged and phase-averaged velocities and turbulent stresses are obtained and they are compared with the experimental data. To investigate mesh dependence on computational results of the LES, two kinds of grid resolution are used. In addition, the effect of a second-order upwind scheme QUICK for convection terms is also investigated due to its dependence on grid resolution. The drag coefficients, the base pressure coefficients and Strouhal numbers are in fairly good agreement with the experimental results, while the computational results show that the artificial dissipative and dispersive effect of QUICK is large in the vicinity of the cylinder in our computation. Thus, it is necessary to use higher-order upwind schemes to reduce the numerical errors, since it is effective in applying LES to practical engineering problems with complicated geometry.     

Basic function scheme of polynomial type

A new numerical method named as basic function method is proposed. It can directly discretize differential operators on unstructured grids. By expanding the basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as a basic function and applying the flux splitting method and the combination of central and upwind schemes to suppress non-physical fluctuation near shock waves, a second-order basic function scheme of polynomial type is proposed to solve inviscid compressible flows numerically. Numerical results of typical examples for two-dimensional inviscid compressible transonic and supersonic steady flows indicate that the new scheme has high accuracy and high resolution for shock waves. Combined with the adaptive remeshing technique, satisfactory results can be obtained.