Viscous Flow Computations Research Articles

Calculation of viscous and inviscid gas flows on unstructured grids using the AUSM scheme

An approach to the solution of the two-dimensional Navier-Stokes equations on triangular unstructured grids is considered. The method is based on the key idea of the Godunov scheme, namely, the advisability of solving the Riemann problem of arbitrary discontinuity breakdown. In the calculations the derivatives with respect to space are approximated with both the first and the second order. However, as distinct from the conventional Godunov method, in calculating the fluxes across the cell boundaries the Riemann problem is solved using the Advection Upstream Splitting Method (AUSM). The concepts involved in the AUSM scheme are discussed. The solution of the discontinuity breakdown problem obtained within the framework of this approach is compared with the results obtained using the Godunov method. Numerical solutions of some problems of viscous and inviscid perfect-gas flows obtained on unstructured grids of different fineness and those obtained on structured grids are also compared. The effect of the spatial approximation order on the accuracy of numerical solutions is studied.

Study of separation phenomenon in transonic flows produced by interaction between shock wave and boundary layer

For compressible flows, the transonic state depends on the geometry, Mach number and the incidence. This effect can produce shock wave. Some studies showed that the interaction between shock wave and boundary layer concerns separation phenomenon. Studies in this report demonstrate conditions of separation in transonic flow and that it is not any interaction between shock wave and boundary layer which can cause boundary layer separation. The studies also show that maximum Mach number in the local supersonic region is not a unique factor influencing the separation, and the separation in transonic flows can occur at the incidence of 0\(^{\circ}\). For the calculation of viscous transonic flows, we use Fluent software with serious treatment of application operation based on the physical nature of phenomenon and the technique of numerical treatment. For the calculation of invicid transonic flows, we built a code solving the full potential equation with verification for accuracy. Results calculated from Fluent have been seriously compared with results of present program and published results in order to assure the accuracy of application operation in the domain of investigation. separation in transonic flows; shock wave and boundary layer

Adaptive Cartesian grid with mesh-less zones for compressible flow calculations

A hybrid grid method that combines the computational efficiency of adaptive Cartesian grids with the flexibility of mesh-less scheme is developed for calculation of compressible flows. The mesh-less zone is created around the geometry by producing layers of nodes along normal direction vectors. Last layer is used as virtual geometry for automatic generation of unstructured Cartesian grid around mesh-less zone. An efficient explicit central difference scheme with artificial dissipation terms and convergence acceleration techniques is developed for both Cartesian grid and mesh-less zones. The method is used for computation of inviscid and viscous flows around single and multi-element airfoils at transonic flow conditions. Results indicate good agreements with those of available unstructured finite-volume flow solver ( Uns2D) and other reference numerical data. The new method is shown to reduce the computational cost up to 70% compared with the fully unstructured flow solver.

A hybrid time-stepping method of local preconditioning on high aspect ratio grids

Time-derivative preconditioning methods have provided robust convergences and accurate solutions for low-speed inviscid flows by using local flow properties as the reference scaling parameter in the preconditioning matrix. During viscous or turbulent flow calculations, the use of high aspect ratio grids to obtain accurate solutions near the wall often results in stiff convergence and nonphysical solutions in the singular regions. In this paper, a switching method is proposed as a remedy to that problem by combining the advantages of the typical local time-stepping and min-CFL/max-VNN methods. The proposed method is based on results of our study of the inviscid bump flow and flat plate boundary layer flows and also proved to be effective to a turbulent flow simulation around the NACA 4412 airfoil.

Multigrid Acceleration and Chimera Technique for Viscous Flow Past a Hovering Rotor

ACCURATE simulation of the viscous flow around a helicopter rotor is still a challenging problem. To predict the blade loading accurately, the numerical method must have a capability to capture the vortical wake generated from the rotor blade tip precisely, which significantly affects the aerodynamic performance of the rotor. Therefore, both the numerical scheme with low numerical dissipation and sufficient dense mesh far away from the blade surface are required for rotor computations to simulate the tip vortex with minimal distortion. Furthermore, for a hovering case, a large numerical integration time is needed for the capture of the fully developed vortical wake [1]. In addition, the existence of incompressible flow near the root will also make the convergence slow. Compared with fixed-wing computations, hovering rotor computations not only require more accurate numerical scheme but also will take much more run time to converge. During the past two decades many numerical methods [2–7] have been developed to simulate the viscous flow around a hovering rotor. The methods [2,3] with the single-block structured grid suffer from the grid-quality problem, because the mesh must be stretched and skewed to cover the complete computational domain and to match the periodic boundary condition. To avoid this problem, the chimera technique [4–7] suggested by Benek et al. [8] was introduced into hovering rotor computations. To capture the tip vortex without unphysical distortion and predict the blade loading accurately, some high-order schemes and vortex-adapted chimera methods [6,7] were presented. Although significant advances in the accurate simulation of the tip vortex have been made with these advanced methods, the large computation cost and the low robustness, as well as the complexity make them far away from engineering applications. As mentioned above, hovering rotor computations, especially the computations with high-order methods, require more run time to converge than their fixed-wing counterparts. A few attempts [1,9,10] have been made to use multigrid methods to accelerate the convergence of hovering rotor calculations. Up to now, there is few works concerning the multigrid method for the viscous flow computation about a hovering rotor based on overset grids. The main objective of the present study is to develop an efficient multigrid algorithm coupled with chimera grids for Navier–Stokes computations about a hovering rotor. It should be mentioned that the experience of this work can be extended to accelerate the computation about a rotor in forward flight using moving overset grids,which is themainmotivation of this research. Furthermore, this experience can also be the basis of the future multigrid computation using high-order methods and chimera grids for rotor flows.

On Resistance Calculation for Autonomous Underwater Vehicles

Investigation on the turbulence model for resistance calculation for autonomous underwater vehicles (AUV) with the typical Myring shape is presented in this paper using computational fluid dynamics (CFD) method. Resistance calculations of the 3D viscous flow over an AUV model are made by solving RANS equations with different viscous models. Comparison with experiments indicates that the SST k-ω two-equation viscous model is the most appropriate model for the resistance prediction.

Two-dimensional compact finite difference immersed boundary method

We present a compact finite differences method for the calculation of two-dimensional viscous flows in biological fluid dynamics applications. This is achieved by using body-forces that allow for the imposition of boundary conditions in an immersed moving boundary that does not coincide with the computational grid. The unsteady, incompressible Navier–Stokes equations are solved in a Cartesian staggered grid with fourth-order Runge–Kutta temporal discretization and fourth-order compact schemes for spatial discretization, used to achieve highly accurate calculations. Special attention is given to the interpolation schemes on the boundary of the immersed body. The accuracy of the immersed boundary solver is verified through grid convergence studies. Validation of the method is done by comparison with reference experimental results. In order to demonstrate the application of the method, 2D small insect hovering flight is calculated and compared with available experimental and computational results. Copyright © 2009 John Wiley & Sons, Ltd.

Viscous flows in corner regions: Singularities and hidden eigensolutions

AbstractNumerical issues arising in computations of viscous flows in corners formed by a liquid–fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the ‘moving contact‐line problem’ and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite‐element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh‐dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh‐independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite‐element method in the context of the present problem are discussed. Copyright © 2009 John Wiley & Sons, Ltd.

A Preconditioned Method for Rotating Flows at Arbitrary Mach Number

An improved preconditioning is proposed for viscous flow computations in rotating and nonrotating frames at arbitrary Mach numbers. The key to the current method is the use of both free stream Mach number and rotating Mach number to construct a preconditioning matrix, which is applied to the compressible governing equations written in terms of primitive variables. A Fourier analysis is conducted that reveals the efficacy of the modified preconditioning. Numerical approximations for the convective and diffusive fluxes are detailed based on the preconditioned system of equations. A set of boundary conditions using characteristic variables are described for internal and external flow computations. Numerical validations are performed on four realistic viscous flows in both fixed and rotating frames. The results indicated that the modified preconditioning has a superior performance compared to the original method to predict flows from extremely low to supersonic Mach numbers.

Solution of 2D Navier–Stokes equation by coupled finite difference-dual reciprocity boundary element method

Computation of viscous fluid flow is an area of research where many authors have tried to present different numerical methods for solution of the Navier–Stokes equations. Each of these methods has its own advantages and weaknesses. In the meantime, many researchers have attempted to develop coupled numerical algorithms in order to save storage for computational purposes and to save computational time. In this paper, a new coupled method is presented for the first time by combining FDM and DRBEM for solving the stream function–vorticity formulation of the Navier–Stokes equations. The vorticity transport equation is analyzed using a finite difference technique while the stream function Poisson’s equation is solved using a dual reciprocity boundary element method. Finally, the robustness and accuracy of the coupled FDM–DRBEM model is proved using the benchmark problem of the flow in a driven square cavity.

The Dynamics of Combustion Products' Flow in a Ring Nozzle with a Half-Closed Cavity

Complex results are presented in numerical and experimental flow dynamic research and thrust characteristics of jet engine model thrust device with ring nozzle working on equilibrium combustion products of acetylene-air mixtures. Experiments with thrust device models were performed in a pulse aerodynamic laboratory facility. Calculations of viscous flow were based on a program complex including the programs of Navier-Stokes equations and numerical integration for various models of gas environment added by the equations of chemical kinetics. As a result, the main details of flow dynamic, stationary flow structure, and thrust characteristics of the ring nozzle have been established. The comparison of experimental and calculated data has been made to verify the mathematical models and computation method.

Implicit fully mesh-less method for compressible viscous flow calculations

A dual-time implicit mesh-less scheme is developed for solution of governing viscous flow equations. The computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing. The Taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization. The capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions. Results are presented which indicate good agreements with the alternative numerical and experimental data. The computational time is considerably reduced when using the proposed mesh-less method compared with the explicit mesh-less and finite-volume counterparts using the same distribution of points.

Combination of adaptive grid‐embedding and equation adaptation methods for compressible viscous flows

AbstractVarious adaptation techniques for the computation of viscous flows have been employed. This work describes the combination of adaptive grid‐embedding and equation adaptation methods on semi‐structured grids for compressible viscous flows. In this combination, Navier–Stokes equations are solved within appreciably viscous macro‐cells, whereas for the remaining macro‐cells the equations are reduced to the Euler equations. Meanwhile, subdividing is used in macro‐cells from the beginning of the adaptation procedure without any restriction. This combination is used to solve three internal and external flow model problems. The combination of adaptive grid‐embedding and equation adaptation is complex, but it is shown in this paper that it is efficient and worth to be used in CPU time‐ and memory space‐consuming problems. Copyright © 2008 John Wiley & Sons, Ltd.

A Three Dimensional Gas-Kinetic Scheme with Moving Mesh for Low-Speed Viscous Flow Computations

The paper introduces the gas-kinetic scheme for three-dimensional (3D) flow simulation. First, under a unified coordinate transformation, the 3D gas- kinetic BGK equation is transformed into a computational space with arbitrary mesh moving velocity. Second, based on the Chapman-Enskog expansion of the kinetic equation, a local solution of gas distribution function is constructed and used in a finite volume scheme. As a result, a Navier-Stokes flow solver is devel- oped for the low speed flow computation with dynamical mesh movement. Several test cases are used to validate the 3D gas-kinetic method. The first example is a 3D cavity flow with up-moving boundary at Reynolds number 3200, where the peri- odic solutions are compared with the experimental measurements. Then, the flow evolution inside a rotating 3D cavity is simulated with the moving mesh method, where the solution differences between 2D and 3D simulation are explicitly pre- sented. Finally, the scheme is applied to the falling plate study, where the unsteady plate tumbling motion inside water tank has been studied and compared with the experimental measurements. AMS subject classifications: 76D05

A high-order gas-kinetic Navier–Stokes flow solver

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge–Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge–Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier–Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier–Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier–Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.

Finite volume numerical simulation of viscoelastic flows in general orthogonal coordinates

This work reports on numerical simulations of unsteady two- and three-dimensional viscoelastic flows in complex geometries. The Navier–Stokes solver is designed for general orthogonal grids in two dimensions and a Cartesian description in the spanwise direction. The conservation equations are written in primitive pressure–velocity variables, making use of the physical curvilinear lengths and contravariant velocities [S.B. Pope, The calculation of turbulent recirculating flows in general orthogonal coordinates, J. Comp. Phys. 26 (1978) 197–217]. The space discretization is finite volume, with the pressure at the center of the control volumes and the velocity components staggered at the center of the cell faces. The conservative advective terms are discretized with a quadratic upwind scheme (QUICK). The extra advective terms induced by coordinate curvature are treated explicitly as integral volumes evaluated at the cell centers. The time advancement of the solution follows from an explicit decoupling procedure [F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965) 2182–2189]. The Adams–Bashforth level 2 scheme is used to evaluate advection, curvature and viscous terms. This discretization produces a symmetric and positive definite matrix for the pressure. The overall formulation is second order accurate in space, first-order in time. To model the viscoelastic flow, the Phan-Thien–Tanner (PTT) constitutive equation is considered. Results are presented for 2D and 3D configurations (flow past a cylinder and flow through a U-curved channel), both presenting recirculations and secondary flows. This contribution demonstrates the versatility of the formulation for 2D and 3D flows with curved boundaries, and its easy implementation starting from a Cartesian description.

The method of lines for the computation of incompressible viscous flows

AbstractA common approach to finding numerical solutions of the time‐dependent incompressible Navier‐Stokes equations is considered within the method of lines framework [1]. For the determination of viscous incompressible flows the stream‐function formulation for the fourth‐order equation [2, 3], an artificial compressibility method [4], and a modified velocity correction method [5] are designed. Some improved and extended results of numerical simulations obtained by the author in the previous works are presented. Test calculations have been done for various flows inside square, triangular and semicircular cavities with one moving wall, the backward‐facing step, double bent channels and for the flow around an aerofoil at large angle of attack. An alternative and practical methodology for resolving the Navier‐Stokes equations in arbitrarily complex geometries using Cartesian meshes is proposed. Some of complex geometrical configurations can be decomposed into a set of subdomains. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computation of incompressible viscous flows in complex geometries by a velocity correction method.

AbstractA method for simulation of viscous flows in complex geometries by resolving the Navier‐Stokes equations with the use of velocity correction procedure is developed. Its virtue lies in the simple description of the flows in terms of primitive variables, such as velocities and pressure, and in its algorithmic simplicity. Decoupling the pressure and velocity reduces the problem to the solution of a sequence of simpler problems. With this method some auxiliary velocity field is found and the projection into the divergence‐free space via the solution of a Poisson equation for some form of the pressure is performed. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Design of multimode axisymmetric hypersonic nozzles with the use of optimization methods

On the basis of the solution of direct problems with the use of various medium models and numerical meth- ods of integration of gas flow equations, methods for designing super- and hypersonic nozzles have been de- veloped. Multimode nozzles in the ranges of Mach numbers 8-14 and 14-20 satisfying given conditions have been designed. optimization. Introduction. It is known that good quality of flow in a wind tunnel is provided by the application of thor- oughly shaped nozzles accelerating the working gas to a given velocity. Such nozzles have a spatial structure, are made with a high precision, and are rather expensive. Practically in all existing wind tunnels single-mode nozzles de- signed for definite operating conditions are used. Because of the complex structure of single-mode nozzles and the high cost of their production, the number of modes of operation of a wind tunnel is always limited. There exist two approaches to the design of multimode shaped nozzles in which the main outlet section has a fixed geometry and the small separable part adjoining the minimal section makes it possible to vary the Mach number at the outlet from the nozzle. The first approach is based on constructing the nozzle contour with the use of the method of characteristics (1-4), and the second approach is based on methods of direct numerical optimization. For example, in (5) the func- tional equal to the sum of standard deviations of the Mach number from its average values in a given region of the flow was minimized. Both approaches did not take into account the presence of viscosity and, accordingly, of the boundary layer that radically changes the characteristics of hypersonic nozzles (6). Therefore, to solve the design prob- lem stated, we used a complex approach combining the method of characteristics that permits calculating fairly easily supersonic nozzles with a uniform outlet characteristic for a perfect nonviscous non-heat-conducting gas, direct meth- ods based on the calculation of viscous flows, and methods of numerical optimization. Construction of the Initial Approximation. The first stage of designing a nozzle consists of constructing by the method of characteristics the nozzle contour for the maximum value of the Mach number at the outlet and deter- mining the size of the acceleration part of the nozzle for the minimum Mach number. This makes it possible to divide the contour into a fixed equailizing part and a variable accelerating part (much smaller in size) whose shape is deter- mined for the chosen value of the Mach number at the outlet from the nozzle (7). This stage is an important element of the chosen strategy since the solution of optimization problems for viscous flows required large computational re- sources and is successful, as a rule, only given a good initial approximation. Design of Multimode Nozzles. Since the viscous flow in a hypersonic nozzle of a given geometry was sup- posed to be calculated by means of the Fluent package, it was necessary to test the accuracy of the solution of the Navier-Stokes equation with the k-ω-model of turbulence. The contour obtained by the method of (5) was used in creating a real axisymmetric nozzle with a nozzle outlet diameter of 0.40 m meant for obtaining Mach numbers 8, 10, 12, and 14 at the outlet. The comparison between the experimental and numerical distributions of the Mach number

The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties

This paper concerns the development of a new Cartesian grid/immersed boundary (IB) method for the computation of incompressible viscous flows in two-dimensional irregular geometries. In IB methods, the computational grid is not aligned with the irregular boundary, and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary, the so-called “cut-cells”. In this paper, we present a new IB method, called the LS-STAG method, which is based on the MAC method for staggered Cartesian grids and where the irregular boundary is sharply represented by its level-set function. This implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut-cells. We have achieved a novel discretization of the fluxes in the cut-cells by enforcing the strict conservation of total mass, momentum and kinetic energy at the discrete level. Our discretization in the cut-cells is consistent with the MAC discretization used in Cartesian fluid cells, and has the ability to preserve the five-point Cartesian structure of the stencil, resulting in a highly computationally efficient method. The accuracy and robustness of our method is assessed on canonical flows at low to moderate Reynolds number: Taylor–Couette flow, flows past a circular cylinder, including the case where the cylinder has forced oscillatory rotations. Finally, we will extend the LS-STAG method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free-stream.