Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).     

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.     

A shock-fitting technique for 2D unstructured grids

A new floating shock-fitting technique featuring the explicit computation of shocks by means of the Rankine-Hugoniot relations has been implemented on two-dimensional unstructured grids. This paper illustrates the algorithmic features of this original technique and the results obtained in the computation of the hypersonic flow past a circular cylinder and a steady Mach reflection.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Optimal control of cylinder wakes via suction and blowing

A general method based on adjoint formulation is discussed for the optimal control of distributed parameter systems (including boundary parameter) which is especially suitable for large dimensional control problems. Strategies for efficient and robust implementation of the method are described. The method is applied to the problem of controlling vortex shedding behind a cylinder (through suction/blowing on the cylinder surface) governed by the unsteady two-dimensional incompressible Navier-Stokes equations space discretized by finite-volume approximation with time-dependent boundary conditions. Three types of objective functions are considered, with regularization to circumvent ill-posedness. These objective functions involve integration over a space-time domain. The minimization of the cost function uses a quasi-Newton DFP method.A complete control of vortex shedding is demonstrated for Reynolds numbers up to 110. The optimal values of the suction/blowing parameters are found to be insensitive to initial conditions of the model when the time window of control is larger than the vortex shedding period, the inverse of the Strouhal frequency. Although this condition is necessary for robust control, it is observed that a shorter window of control may suffice to suppress vortex shedding.     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

基于结构网格的高超声速流动大规模并行数值模拟研究

本文采用MPI消息传递模式自主开发出适用于高超声速流动数值模拟的并行计算软件,该软件以三维Navier-Stokes方程为基本控制方程来求解层流问题,应用基于结构网格的有限体积法对计算域进行离散,采用AUSMPW+格式求解对流通量,利用MUSCL插值方法获得高阶精度,时间格式上采用LU-SGS方法进行时间迭代以加快求解定常流动的收敛过程。在高性能计算机上针对不同高超声速流动进行大规模并行计算的结果表明,所开发的CFD并行计算软件具有较高的并行计算效率,为高超声速飞行器气动力/热的准确预测提供了高效工具。     

Parallel three-dimensional direct simulation Monte Carlo method and its applications

The development of a parallel three-dimensional direct simulation Monte Carlo (DSMC) method using unstructured cells is reported. Variable hard sphere molecular model and no time counter method are used for the molecular collision kinetics, while the cell-by-cell ray-tracing technique is implemented for particle movement. Developed serial code has been verified by comparing the results of a supersonic corner flow with those of Bird’s three-dimensional structured DSMC code. In addition, a benchmark test is performed for an orifice expanding flow to verify the parallel implementation of DSMC method by comparing with available experimental data. Static physical domain decomposition is used to distribute the workload among multiple processors by considering the estimated particle weighting distribution. Two-step multi-level graph partitioning technique is used to perform the required domain decomposition. Completed code is then applied to compute a hypersonic flow over a sphere (external flow) and the flow field of a spiral drag pump (internal flow), respectively. Results of the former are in good agreement with previous numerical results using axisymmetric DSMC method and experimental data. Results of the latter also agree well with previous numerical results.     

DNS for flow separation control around an airfoil by pulsed jets

Direct numerical simulation (DNS) for flow separation and transition around a NACA-0012 airfoil with an attack angle of 4° and Reynolds number of 100,000 has been reported in our previous paper. The details of flow separation, formation of the detached shear layer, Kelvin-Helmholtz instability (inviscid shear layer instability) and vortex shedding, interaction of nonlinear waves, breakdown, and re-attachment are obtained and analyzed. The power spectral density of pressure shows the low frequency of vortex shedding caused by the Kelvin-Helmholtz instability still dominates from the leading edge to trailing edge. Based on our understanding on the flow separation mechanism, we try to reveal the mechanism of the flow separation control using blowing jets and then optimize the jets. DNS simulations for flow separation control by blowing jets (pulsed and pitched and skewed jets) are reported and analyzed. The effects of different unsteady blowing jets on the surface at the location just before the separation points are studied. The length of separation bubble is significantly reduced (almost removed) after unsteady blowing technology is applied. The mechanism of early transition caused by the blowing jets was found. A blowing jet with K-H frequency, sharp shape function (very small mass blowing), pitching and skewing obtained the best efficiency based on the increase of the ratio of lift over drag and decrease of blowing mass flow. In this work, a DNS code with high-order accuracy and high-resolution developed by the computational fluid dynamics group at University of Texas at Arlington is applied.     

A numerical investigation of fluid flow over a rotating cylinder with cross flow oscillation

This paper studies a two-dimensional incompressible viscous flow past a rotating cylinder with cross flow oscillation using a finite element method based on the characteristic based split (CBS) algorithm to solve governing equations including full Navier–Stokes and continuity equations. Dynamic unstructured triangular grid is used employing lineal and torsional spring analogy which is coupled with the solver by an Arbitrary Lagrangian–Eulerian (ALE) formulation. After verifying the accuracy of the numerical code, simulations are conducted for the flow past a rotating cylinder with cross flow oscillation at moderate Reynolds numbers of 50, 100, and 200 considering different non-dimensional rotational speeds based on the free-stream velocity in the range 0–2.5, and various oscillating amplitudes and frequencies. Effects of the oscillation and rotation of the cylinder on the vortex shedding both in lock-on and non-lock-on regions, the mean drag and lift coefficients, and the Strouhal number are investigated in detail. It is found that similar to the fixed cylinder beyond a critical non-dimensional rotational speed the vortex shedding is highly suppressed. In addition, by increasing the rotational speed of the cylinder, the lift coefficient increases while decreasing the drag coefficient. However, in the vortex lock-on region both the lift and the drag coefficients increase significantly.     

Numerical prediction of unsteady vortex shedding for large leading-edge roughness

A full two-dimensional Navier-Stokes algorithm is used to investigate unsteady, incompressible viscous flow past an airfoil leading edge with surface roughness that is characteristic of ice accretion. The roughness is added to the surface through the use of a Prandtl transposition and can generate both small-scale and large-scale roughness. The focus of the study is a detailed flow analysis of the unsteady velocity fluctuations and vortex shedding induced by the surface roughness. The results of this study are compared to experimental data on roughness-induced transition for the same roughness geometry. A comparison is made between “fluctuation intensity” values from the current algorithm to experimentally determined turbulence intensity values. The effects of the roughness Reynolds number, Re_k, are investigated and compared to experimental values of the critical roughness Reynolds number. The authors speculate that there may be a possible correlation between unsteady roughness-induced vortex shedding and the onset of experimentally measured transitional flow downstream of large-scale roughness.     

An open source, parallel DSMC code for rarefied gas flows in arbitrary geometries

This paper presents the results of validation of an open source Direct Simulation Monte Carlo (DSMC) code for general application to rarefied gas flows. The new DSMC code, called dsmcFoam, has been written within the framework of the open source C++ CFD toolbox OpenFOAM. The main features of dsmcFoam code include the capability to perform both steady and transient solutions, to model arbitrary 2D/3D geometries, and unlimited parallel processing. Test cases have been selected to cover a wide range of benchmark examples from 1D to 3D. These include relaxation to equilibrium, 2D flow over a flat plate and a cylinder, and 3D supersonic flows over complex geometries. In all cases, dsmcFoam shows very good agreement with data provided by both analytical solutions and other contemporary DSMC codes.     

Numerical simulation of flow past row of square cylinders for various separation ratios

A systematic study for the flow around a row of five square cylinders placed in a side-by-side arrangement and normal to the oncoming flow at a Reynolds number of 150 is carried out through the numerical solution of the two-dimensional unsteady incompressible Navier-Stokes equations. Special attention is paid to investigate the effect of the spacing between the five cylinders on the wake structure and vortex shedding mechanism. The simulations are performed for the separation ratios (spacing to size ratio) of 1.2, 2, 3 and 4. Depending on the separation ratio the following flow patterns are observed: a flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. These flow patterns are supposed to be a consequence of the interaction between two types of frequencies viz. the vortex shedding (primary) and the cylinder interaction (secondary) frequencies. At small separation ratio the flow is predominantly characterized by the jet in the gaps between successive cylinders and the secondary frequencies play a role in the resulting chaotic flow. On the contrary, at higher separation ratio the secondary frequencies almost disappear and the resulting flow becomes more synchronized dominated by the primary frequency.     

A numerical analysis of two-dimensional flow past a rectangular prism by a discrete vortex model

A discrete vortex model is developed to analyse the two-dimensional fully separated unsteady flow past a rectangular prism. The effects of viscous diffusion of vortices and the loss in vorticity after the stationary prism with a thickness ratio ranging from 0.5 to 2.0. The formation of Karman vortices, the vortex shedding frequency, and the fluid forces acting on the body are favourably compared with the experimental results by various research workers. The method of analysis is also shown to be applicable to the flow past a prism that is in forced vibration.     

Contrast between steady and time-averaged unsteady combustion simulations

Time-averaged predictions from unsteady solutions of the two-dimensional Navier–Stokes equations are contrasted with Reynolds-averaged results for a reacting flow problem in a high pressure combustor. The goal is to determine whether the two-dimensional unsteady approximation can be useful as an engineering analysis in problems for which time-averaged quantities are of primary interest. The conditions are taken from an experiment in which non-premixed gaseous oxygen and hydrogen were injected into a combustion chamber through coaxial channels. The resulting flowfield is dominated by a large recirculation zone arising from the back-step created by the injector. The results of steady and time-averaged, unsteady solutions are strikingly different. The unsteady simulation produces strong unsteady structures whose time-averaged results lead to a much wider flame zone, a different recirculation zone structure, and a substantially different wall heat flux than those obtained with a steady RANS procedure. The time-averaged calculations yield the correct combustor chamber pressure and compare considerably more favorably with heat flux measurements than do the RANS results. The two-dimensional approximation, however, overstates the unsteady vortex roll up and precludes large scale mixing across the axis of symmetry, thereby giving deficient predictions near the centerline. Overall, the present results indicate that capturing large-scale unsteady characteristics can provide more accurate predictions of recirculation dominated reacting flows and suggest that two-dimensional, time-averaged solutions represent a potentially useful engineering tool for problems of this nature while also serving as a precursor for full three-dimensional simulations.     

Surface effects on transient three-dimensional flows around rotating spheres at moderate Reynolds numbers

Transient wake flow patterns and dynamic forces acting on a rotating spherical particle with non-uniform surface blowing are studied numerically for Reynolds numbers up to 300 and dimensionless angular velocities up to Ω=1. This range of Reynolds numbers includes the three distinct wake regimes i.e., the steady axisymmetric, the steady non-symmetrical and the unsteady with vortex shedding. The Navier–Stokes equations for an incompressible viscous flow are solved by a finite volume method in a three-dimensional, time accurate manner. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that unsteady three-dimensional wake flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects and thus, vortex shedding occurs even at Re=200. Global properties, such as the lift and drag coefficients, and the Strouhal number are also significantly affected. It is shown that the present data for the average lift and drag coefficients correlate well with:

C_L/(1+Ω)^3.6=0.11

C_D(1+20V_S)^0.2/(1+Ω)^Re/1000=24(1+Re^2/3/6)/Re

where V_S is the average surface blowing velocity normalized by the free stream velocity.     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

Flow characteristics of two rotating side-by-side circular cylinder

This paper presents a numerical investigation of the characteristics of the two-dimensional laminar flow around two rotating circular cylinders in side-by-side arrangements. In order to consider the combined effects of the rotation and the spacing between two cylinders on the flow, numerical simulations are performed at a various range of absolute rotational speeds (|α|?2) for four different gap spacings of 3, 1.5, 0.7 and 0.2 at Reynolds number of 100 showing the typical two-dimensional vortex shedding. As |α| increases, the flow changes its condition from periodic to steady after a critical rotational speed, which depends on the gap spacing. In the cases of gap spacings of 3 and 0.2, the wake keeps the same pattern, until flow reaches the steady state. However, for the gap spacings of 1.5 and 0.7, the wake patterns change in the unsteady regimes. For the cases in which the flow is unsteady, the Strouhal number strongly depends on the gap. For a fixed gap spacing, the variation of the Strouhal number is significant when the wake pattern is changed according to the rotational speed. Regardless of the gap spacing, as |α| increases, the lift increases and the drag decreases. Quantitative information about the flow variables such as the pressure coefficient and wall vorticity distributions on the cylinders is highlighted.     

End-wall effects on vortex shedding in planar shear flow over a circular cylinder

A prevailing controversy regarding the suppression of periodic vortex shedding from a circular cylinder embedded in a planar shear flow has been addressed. Three-dimensional computer simulations utilizing the advanced MGLET software [11] demonstrated the importance of the end-wall conditions. Earlier results from two-dimensional simulations at Re = 100 were reproduced only with free-slip conditions. With no-slip conditions imposed at one or both end-walls, the vortex shedding was suppressed near the no-slip boundary and the shedding pattern was substantially affected even at mid-span. The Strouhal number decreased when the shear-rate parameter was increased from 0.1 to 0.2, irrespective of the choice of boundary conditions.