Viscous Flow Computations Research Articles

Bracket formulations and energy- and helicity-preserving numerical methods for the three-dimensional vorticity equation

The vorticity equation for three-dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the Poisson or Nambu bracket together with a dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method. In particular, energy and helicity are conserved precisely in inviscid flow computations. The energy and enstrophy dissipate properly owing to viscosity in viscous flow computations, and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations. The relationships between the stream function, velocity, and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited. Numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method.

Explicit discontinuous spectral element method with entropy generation based artificial viscosity for shocked viscous flows

A spatio-temporal adaptive artificial viscosity (AV) based shock-capturing scheme is proposed for the solution of both inviscid and viscous compressible flows using a high-order parallel Discontinuous Spectral Element Method (DSEM). The artificial viscosity and artificial thermal conduction coefficients are proportional to the viscous and thermal entropy generating terms, respectively, in the viscous entropy conservation law. The magnitude of AV is limited based on the explicit stable CFL criterion, so that the stable artificial viscous time step size is greater than the convective stable time step size. To further ensure the stability of this explicit approach, an adaptive variable order exponential filter is applied, if necessary, in elements where the AV has been limited. In viscous flow computations a modified Jameson's sensor (Ducros et al., 1999 [61]) limits the AV to small values in viscous shear regions, so as to maintain a high-order resolution in smooth regions and an essentially non-oscillatory behavior near sharp gradients/shocks regions. We have performed a systematic and extensive validation of the algorithm with one-dimensional problems (inviscid moving shock and viscous shock-structure interaction), two-dimensional problems (inviscid steady and unsteady shocked flows and viscous shock-boundary layer interaction), and a three-dimensional supersonic turbulent flow over a ramped cavity. These examples demonstrate that the explicit DSEM scheme with adaptive artificial viscosity terms is stable, accurate and efficient.

A pressure-based Mach-uniform method for viscous fluid flows

ABSTRACTA pressure-based, Mach-uniform method is developed by combining the SLAU2 numerical scheme and the higher temporal order pressure-based algorithm. This hybrid combination compensates the limitation of the SLAU2 numerical scheme in the low-Mach number regime and deficiencies of the pressure-based method in the high-Mach number regime. A momentum interpolation method is proposed to replace the Rhie-Chow interpolation for accurate shock-capturing and to alleviate the carbuncle phenomena. The momentum interpolation method is consistent in addition to preserving pressure–velocity coupling in the incompressible limit . The postulated pressure equation allows the algorithm to compute the subsonic flows without empirical scaling of numerical dissipation at low-Mach number computation. Several test cases involving a broad range of Mach number regimes are presented. The numerical results demonstrate that the present algorithm is remarkable for the calculation of viscous fluid flows at arbitrary Mach number including shock wave/laminar boundary layer interaction and aerodynamics heating problem.

Kinetic energy preserving DG schemes based on summation‐by‐parts operators on interior node distributions

AbstractIn this work, a kinetic energy preserving DG scheme in one space dimension using Gauss‐Legendre nodes is presented. Stability problems will be demonstrated when using interface terms that are derived from the Lobatto nodes within the Gauss‐Legendre skew‐symmetric DG formulation. However, combined with correct interface terms, the skew‐symmetric DG scheme constructed on Gauss‐Legendre nodes is expected to yield higher accuracy compared to its Gauss‐Lobatto counterpart. This advantage is demonstrated in the case of viscous compressible flow computation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimization of a twin-skeg container vessel by parametric design and CFD simulations

The model tests results for the original lines of an 10000TEU container vessel show that the delivered power is higher and could not satisfy the requirement of energy saving effects and design targets. In this paper, the lines optimization of the 10,000 twin-skeg container vessel was carried out by parametric modeling and CFD simulations. At first, the CFD methods for twin-skeg hull form were validated by the comparison with the experimental results. Then more than one hundred parameters were adopted for the establishment of the fully parametric model. Based on the parametric model of the twin-skeg container vessel, the preliminary optimization was carried out by tight coupling of FRIENDSHIP-FRAMEWORK with potential flow of SHIPFLOW. Then several important parameters related to the after part of twin-skeg vessel were investigated by viscous flow computation. The final optimized variant PM11, which the total resistance was reduced by about 8.3% in model scale, is obtained within the constraints of general arrangement. And the model tests for variant PM11 was carried out in CSSRC, which shows that the resistance of optimized variant PM11 is decreased by about 8.6%.

Hydrodynamic structure of viscous flows in rotating convergent-divergent channels

The results of the numerical calculations of isothermal viscous flows in convergent-divergent channels axially rotating at a constant angular velocity are presented. With reference to the example of the calculations performed at Re = 2000 the difference in the flowfields in a fixed channel and that rotating at an angular velocity Ω0 = 10 s−1 in analyzed. The results of the calculations of viscous flows in convergent-divergent channels at Re = 2000 and Ω0 = 10 s−1 are also presented.

A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier–Stokes solutions

In this paper, for the first time a third-order compact gas-kinetic scheme is proposed on unstructured meshes for the compressible viscous flow computations. The possibility to design such a third-order compact scheme is due to the high-order gas evolution model, where a time-dependent gas distribution function at cell interface not only provides the fluxes across a cell interface, but also presents a time accurate solution for flow variables at cell interface. As a result, both cell averaged and cell interface flow variables can be used for the initial data reconstruction at the beginning of next time step. A weighted least-square procedure has been used for the initial reconstruction. Therefore, a compact third-order gas-kinetic scheme with the involvement of neighboring cells only can be developed on unstructured meshes. In comparison with other conventional high-order schemes, the current method avoids the Gaussian point integration for numerical fluxes along a cell interface and the multi-stage Runge–Kutta method for temporal accuracy. The third-order compact scheme is numerically stable under CFL condition CFL≈0.5. Due to its multidimensional gas-kinetic formulation and the coupling of inviscid and viscous terms, even with unstructured meshes, the boundary layer solution and vortex structure can be accurately captured by the current scheme. At the same time, the compact scheme can capture strong shocks as well.

A multi-dimensional high-order DG-ALE method based on gas-kinetic theory with application to oscillating bodies

This paper presents a multi-dimensional high-order discontinuous Galerkin (DG) method in an arbitrary Lagrangian-Eulerian (ALE) formulation to simulate flows over variable domains with moving and deforming meshes. It is an extension of the gas-kinetic DG method proposed by the authors for static domains (X. Ren et al., 2015 22). A moving mesh gas kinetic DG method is proposed for both inviscid and viscous flow computations. A flux integration method across a translating and deforming cell interface has been constructed. Differently from the previous ALE-type gas kinetic method with piecewise constant mesh velocity at each cell interface within each time step, the mesh velocity variation inside a cell and the mesh moving and rotating at a cell interface have been accounted for in the finite element framework. As a result, the current scheme is applicable for any kind of mesh movement, such as translation, rotation, and deformation. The accuracy and robustness of the scheme have been improved significantly in the oscillating airfoil calculations. All computations are conducted in a physical domain rather than in a reference domain, and the basis functions move with the grid movement. Therefore, the numerical scheme can preserve the uniform flow automatically, and satisfy the geometric conservation law (GCL). The numerical accuracy can be maintained even for a largely moving and deforming mesh. Several test cases are presented to demonstrate the performance of the gas-kinetic DG-ALE method.

Viscous-flow Calculations of Submarine Maneuvering Hydrodynamic Coefficients and Flow Field based on Same Grid Topology

Viscous-flow Calculations of Submarine Maneuvering Hydrodynamic Coefficients and Flow Field based on Same Grid Topology

A diffusion-regulated scheme for the compressible Navier–Stokes equations using a boundary-layer sensor

A new formulation to regulate the numerical diffusion of the Local Lax–Friedrichs (LLF) scheme for the accurate computation of high-speed viscous flows is proposed. This is achieved by modifying the diffusion regulation (DR) parameter of the DRLLF scheme originally proposed by Jaisankar and Raghurama Rao for the Euler equations. The modified version of the DR parameter operates based on a boundary layer sensor in such a way that the numerical diffusion is further reduced inside the viscous zone only and in the outer inviscid zone the parameter reverts back to the original inviscid formulation. This new scheme is named as the DRLLF-Viscous (DRLLFV) scheme. Test cases of viscous supersonic flow over a flat plate and hypersonic flow over a ramped surface are computed using the new scheme and compared with the AUSM scheme, DRLLF scheme and van Leer's Flux Vector Splitting (FVS) scheme. Comparisons with the literature show that the DRLLFV scheme resolves the boundary layer including flow separation more accurately than van Leer's FVS and DRLLF schemes and nearly as accurately as the AUSM scheme. In the inviscid zone, the DRLLFV scheme captures the leading-edge shock better than the rest of the schemes. If the overall results in both the viscous and inviscid regions are considered, the new scheme is seen to perform better than the AUSM scheme.

Кінцево-різницевий метод розрахунку течії газу у дозвуковому газовому ежекторі

Проведено аналіз сучасних математичних моделей турбулентної в’язкості для численного моделювання течії в’язкого газу в дозвукових газових ежекторах з поворотом потоку. Розглянуті переваги та недоліки кожного з них. Запропоновано до використання кінцево-різницевий метод для проведення розрахунку течії в’язкого газу в дозвукових газових ежекторах.

Modelling 3D Steam Turbine Flow Using Thermodynamic Properties of Steam Iapws-95

Abstract An approach to approximate equations of state for water and steam (IAPWS-95) for the calculation of three-dimensional flows of steam in turbomachinery in a range of operation of the present and future steam turbines is described. Test calculations of three-dimensional viscous flow in an LP steam turbine using various equations of state (perfect gas, Van der Waals equation, equation of state for water and steam IAPWS-95) are made. The comparison of numerical results with experimental data is also presented.

Effects of Numerical Diffusion on the Computation of Viscous Supersonic Flow Over a Flat Plate

The effects of numerical diffusion on the computation of supersonic viscous flow over a flat plate at zero incidences are numerically investigated. The inviscid flux terms in the Navier–Stokes equations are computed using three schemes, namely, van Leer’s Flux Vector Splitting, Liou and Steffen’s Advection Upstream Splitting Method (AUSM) and Jaisankar and Raghurama Rao’s Diffusion Regulated Local Lax Friedrichs (DRLLF) schemes. The results are correlated with the inherent numerical diffusion of these schemes. The study is also motivated by the necessity to examine whether reduced artificial viscosity can be used with the DRLLF scheme in computing viscous flows than was suggested in the original paper on inviscid computation. It is demonstrated that reduced artificial viscosity is not only possible, but it results in a scheme that is very efficient in the computation of the standard supersonic viscous flow over a flat plate, in that it is comparable in accuracy to the AUSM scheme in the boundary layer and better than it in shock resolution. Even with the reduced artificial viscosity suggested in this paper the DRLLF scheme shows good convergence behaviour comparable with the other two schemes.

A third-order gas-kinetic scheme for three-dimensional inviscid and viscous flow computations

With the WENO reconstruction, a third-order multidimensional gas-kinetic scheme, with inclusion of both normal and tangential derivatives of macroscopic flow variables, is constructed for the three-dimensional inviscid and viscous flow computations. For the flux evaluation, a time and space dependent gas distribution function from an initial piece-wise discontinuous polynomials around a cell interface is presented, where a multiple scale physical process from the kinetic to the hydrodynamic one is used for the flow evolution process. A high-order accuracy is achieved in the current scheme by integrating the space and time dependent flux function over the surface of a cell interface and a whole time step, without using the conventional Gaussian points integration for spatial accuracy and Runge–Kutta method for temporal accuracy, which have been used in many other high-order schemes. This scheme is applied to both inviscid and viscous, and low and high speed flow computations. The numerical tests clearly demonstrate that the current scheme is robust for the flows with strong discontinuities and accurate for viscous smooth flow solutions. The continuous effort on the gas kinetic scheme (GKS) is to develop a third method which has the same reliability and robustness as the well-developed second-order shock capturing schemes, but is simply much more accurate in all flow applications. The current scheme seems achieve such a target.

New high-resolution-preserving sliding mesh techniques for higher-order finite volume schemes

This paper presents a new sliding mesh technique for the computation of unsteady viscous flows in the presence of rotating bodies. The compressible Euler and incompressible Navier–Stokes equations are solved using a higher-order (>2) finite volume method on unstructured grids. A sliding mesh approach is employed at the interface between computational grids in relative motion. In order to prevent loss of accuracy, two distinct families of higher-order sliding mesh interfaces are developed. These approaches fit naturally in a high-order finite volume framework. To this end, Moving Least Squares (MLS) approximants are used for the transmission of the information from one grid to another. A particular attention is paid for the study of the accuracy and conservation properties of the numerical scheme for static and rotating grids. The capabilities of the present solver to compute complex unsteady vortical flow motions created by rotating geometries are illustrated on a cross-flow configuration.

A multi-dimensional high-order discontinuous Galerkin method based on gas kinetic theory for viscous flow computations

This paper presents a high-order discontinuous Galerkin (DG) method based on a multi-dimensional gas kinetic evolution model for viscous flow computations. Generally, the DG methods for equations with higher order derivatives must transform the equations into a first order system in order to avoid the so-called “non-conforming problem”. In the traditional DG framework, the inviscid and viscous fluxes are numerically treated differently. Differently from the traditional DG approaches, the current method adopts a kinetic evolution model for both inviscid and viscous flux evaluations uniformly. By using a multi-dimensional gas kinetic formulation, we can obtain a spatial and temporal dependent gas distribution function for the flux integration inside the cell and at the cell interface, which is distinguishable from the Gaussian Quadrature point flux evaluation in the traditional DG method. Besides the initial higher order non-equilibrium states inside each control volume, a Linear Least Square (LLS) method is used for the reconstruction of smooth distributions of macroscopic flow variables around each cell interface in order to construct the corresponding equilibrium state. Instead of separating the space and time integrations and using the multistage Runge–Kutta time stepping method for time accuracy, the current method integrates the flux function in space and time analytically, which subsequently saves the computational time. Many test cases in two and three dimensions, which include high Mach number compressible viscous and heat conducting flows and the low speed high Reynolds number laminar flows, are presented to demonstrate the performance of the current scheme.

Construction of Godunov-Type Difference Schemes in Curvilinear Coordinates and an Application to Spherical Coordinates

A method is proposed for the construction of conservative flow difference schemes for the calculation of compressible viscous gas flows in curvilinear coordinates based on an arbitrary Godunov-type Cartesian scheme. The realization of the scheme in curvilinear coordinates requires minimal additions to the basic Cartesian scheme code. The method is illustrated in application to the case of spherical coordinate. A spherical scheme of second-order spatial approximation is constructed. Results of test calculations are reported.

Calculating Methods Analysis of Variable Guide Vane Blade System Characteristics

There are presented calculating study of characteristics (loss factor and the lag angle of flow) of variable guide vane blade system for high pressure stage of centrifugal compressor with the influence of blades rotation angles (attack angle). The results are obtained using the known dependencies and using the calculation of viscous flow with ANSYS CFX and they are analyzed.

Unbounded Immersed Interface Solver for Vortex Particle-Mesh Methods

This paper aims at presenting an alternative and more consistent way to account for bodies in the framework of computing incompressible external flows in aerodynamics using vortex particle-mesh methods (VPM). More specifically, we present a purely grid-based numerical technique using an immersed interface method and the James-Lackner algorithm to compute the 2-D potential flow in an unbounded domain, including irregular interior boundaries and also the presence of a given vorticity field. This ingredient is one of the main components that are required for the computation of general viscous flows. It is therefore the first step towards an integrated unbounded immersed interface VPM method.

An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow

This paper presents a novel algorithm to solve the 2-D potential flow past complex geometries with circulation in unbounded domain and in the presence of a given vorticity field. It is based on a Poisson solver that combines two components: the immersed interface method to enforce the boundary condition on each inner boundary and the James–Lackner algorithm to compute the outer boundary condition consistent with the unbounded domain solution. The algorithm is here based on second order finite differences and it requires solely 1-D stencil corrections; this makes the immersed interface part of the present method easily extendable to higher dimensional problems. The treatment of the outer boundaries requires an iterative boundary potential method. The algorithm is validated, by means of grid convergence studies, on the flow past multiple bodies. The results confirm the second order accuracy everywhere. The algorithm is also self consistent as “all is done on the grid” (thus without using a vortex panel boundary element method in addition to the grid). For cusped airfoils, a consistent way to enforce the Kutta–Joukowsky condition is also presented. The present algorithm constitutes a crucial building block towards an immersed interface-enabled vortex particle-mesh method for the computation of unsteady viscous flows, with boundary layers, detached shear layers and wakes. A possible extension to 3-D problems is also briefly discussed.