Roe Scheme Research Articles

Numerical simulation of gas-liquid two-phase flow in wellbore based on drift flux model

A general model and computational code for transient gas-liquid two-phase flow in wellbore are developed in this paper. The model consists of mass conservation equation for each phase and mixture momentum conservation equation. The interactions of each phase are taken into account by invoking the Shi slip relation in the proposed model. The model is numerically solved by the second order AUSMV scheme, which is obtained by using classical MUSCL technique. The computational code is validated by the classical numerical cases, Zuber–Findlay tube and complex mass transport problem. The simulation results show good agreement with the references which are obtained by using first-order flux-limited Roe scheme with refined grids. The computational code also succeeds in obtaining the transient behavior of each phase for a normal gas-kick occurrence in wellbore. The simulation results illustrate the degree of pressure variation in the upper wellbore is greater than that in the lower wellbore. The pressures of wellbore increase before the corresponding gas mass flow rate reach max inlet gas mass flow rate, then decrease.

Coupled finite-volume scheme with adapted Augmented Roe scheme for simulating morphological evolution in arbitrary cross-sections

Abstract A new coupled finite-volume scheme based on the Augmented Roe solver adapted to simulate morphological evolution of arbitrary cross-sections is presented. In pure hydrodynamic conditions, the Augmented Roe scheme has proven to provide accurate results and a constant discharge in steady-flow conditions. Here, this scheme is extended to solve the one-dimensional Saint-Venant–Exner system of equations written for arbitrary cross-sections. Therefore, new eigenvalues and source-term calculations are proposed to account for the irregular shape of the cross-sections. The performances of the proposed scheme are assessed by comparison with three different one-dimensional numerical models aimed at simulating morphological changes, with coupled or uncoupled approaches, and based on HLL or Roe-based flux calculations. Numerous test cases were examined, including water at rest, steady flows and transient flows for which experimental results exist. The results show that the proposed scheme provides stable and accurate results for a wider range of situations than is available with other classical models.

Investigation of boundary layer thickness and turbulence intensity on film cooling with a fan-shaped hole by direct numerical simulation

Effects of the mainstream boundary layer thickness and the turbulence intensity on film cooling under low Reynolds number conditions are studied in this work by the direct numerical simulation (DNS). In other to solve low-speed compressible flow problems, several methods of Roe scheme, preconditioning, dual time stepping, and LUSGS are adopted to solve governing equations. Results reveal that a horseshoe vortex appears with a thicker mainstream boundary layer, and thus the lateral coverage of the coolant fluid has increased significantly. In addition, the existence of turbulence intensity eliminates the blow-off phenomenon, which happens in a thin mainstream boundary layer condition and enhances the film cooling effectiveness.

A New Robust Carbuncle-Free Roe Scheme for Strong Shock

The paper devises a new robust carbuncle-free Roe Riemann solver for strong shock, different from hybrid method, entropy fix, Liou’s conjecture (J Comput Phys 160:623–648, 2000) and artificial viscosity (Rodionov in J Comput Phys 345:308–329, 2017). Roe scheme encounters carbuncle phenomenon, violates entropy condition and lacks positivity property. The remedy integrates shear viscosity into momentum flux to damp undesirable perturbation and prevents unstable vorticity mode from triggering shock instability. Compared to HLL scheme, shear viscosity is properly established through dimensional analysis and analogy method. The non-linear wave speeds are slightly modified by incorporating the neighboring cell information for positive conservation. The pressure-based sensing function is applied to preserve shear layer while keeping shock robustness. The resulting scheme is very easily implemented by converting the existing Roe code. The matrix stability analysis confirms that this approach is shock-stable and more robust than entropy fix. A series of numerical results demonstrate its potential features: positivity-preserving property, entropy-satisfying property, accurate boundary-layer resolution and high robustness against shock instability. In addition, it signifies that the reason to cause shock instability for Roe scheme may be not the pressure difference term in mass flux, but the inadequate shear viscosity.

Direct numerical simulation of film cooling with a fan-shaped hole under low Reynolds number conditions

Shaped film cooling holes have several features that can greatly improve film cooling effectiveness, and it has been studied and utilized in gas turbine engines for decades. Few studies, however, have reported the effects of low mainstream Reynolds number on shaped film cooling holes. In this study, the effects of mainstream Reynolds number on film cooling with a fan shaped hole are studied by utilizing direct numerical simulation (DNS). In addition, the compressibility and the viscosity of the working fluid are simultaneously considered, and the non-reflecting and absorbing boundary conditions are adopted at the exit of the main channel. The methods of the Roe scheme, preconditioning, and dual time stepping are employed together to solve the governing equations of a low-speed compressible flow problem. This study considers the mainstream Reynolds numbers of ReD = 480 and 3200 with 0% and 5% turbulence intensity in the mainstream. Results reveal that the coolant jet penetrates into the mainstream with a mainstream Reynolds number of 480. However, at the higher Reynolds number, the coolant jet develops along the wall and results in better film cooling effectiveness. In addition, special attention is paid to the structures of the vortices developed from the crossflow. Hairpin vortices become smaller at higher mainstream Reynolds numbers. On the contrary, horseshoe vortices appear when the mainstream Reynolds number is increased. A detailed comparison of the vortices is presented in this study.

TVD–MUSCL Finite Volume Model Using in Tidal Bore Simulation Research

Most of the numerical methods used for shallow water flow calculation are not applicable to the numerical calculation of the discontinuous flows. High precision resolution is a difficult problem in simulate discontinuous flows. This paper adopts Roe scheme approximate Riemann solver to deal with normal vector through interface, apply TVD–MUSCL scheme and Hancock scheme to increase the precision of space and time to second order using in tidal bore simulation. In order to verify the accuracy of the model and simulate the ability of discontinuous flows, some numerical examples are simulated in this paper, the results of simulation and other scholars’ results are compared to check the applicability and accuracy of the model for discontinuous flow. With the generalized river course tidal bore experiment to prove the model has the ability to capturing shock wave and better calculative precision.

Density enhancement mechanism of upwind schemes for low Mach number flows

Many all-speed Roe schemes have been proposed to improve performance in terms of low speeds. Among them, the F-Roe and T-D-Roe schemes have been found to get incorrect density fluctuation in low Mach flows, which is expected to be with the square of Mach number. Asymptotic analysis presents the mechanism of how the density fluctuation problem relates to the incorrect order of terms in the energy equation $${{\tilde{\rho }} {\tilde{a}} {\tilde{U}}\varDelta U}$$ . It is known that changing the upwind scheme coefficients of the pressure-difference dissipation term $$D^P$$ and the velocity-difference dissipation term in the momentum equation $$D^{\rho U}$$ to the order of $$O(c^{-1})$$ and $$O(c^{0})$$ can improve the level of pressure and velocity accuracy at low speeds. This paper shows that corresponding changes in energy equation can also improve the density accuracy in low speeds. We apply this modification to a recently proposed scheme, TV-MAS, to get a new scheme, TV-MAS2. Unsteady Gresho vortex flow, double shear-layer flow, low Mach number flows over the inviscid cylinder, and NACA0012 airfoil show that energy equation modification in these schemes can obtain the expected square Ma scaling of density fluctuations, which is in good agreement with corresponding asymptotic analysis. Therefore, this density correction is expected to be widely implemented into all-speed compressible flow solvers.

A Low Cost Numerical Simulation of a Supersonic Wind-tunnel Design

In the present paper, a supersonic wind-tunnel is designed to maintain a flow with Mach number of 3 in a 30cm×30cm test section. An in-house CFD code is developed using the Roe scheme to simulate flow-field and detect location of normal shock in the supersonic wind-tunnel. In the Roe scheme, flow conditions at inner and outer sides of cell faces are determined using an upwind biased algorithm. The in-house CFD code has been parallelized using OpenMp to reduce the computational time. Also, an appropriate equation is derived to predict the optimum number of cores for running the program with different grid sizes. In the design process of the wind-tunnel, firstly geometry of the nozzle is specified by the method of characteristics. The flow in the nozzle and test section is simulated in the next step. Then, design parameters of the diffuser (convergence and divergence angles, area of the throat, and ratio of the exit area to the throat area) are determined by a trial and error method. Finally, an appropriate geometry is selected for the diffuser which satisfies all necessary criteria.

Wave Structure Similarity of the HLLC and Roe Riemann Solvers: Application to Low Mach Number Preconditioning

The approximate Riemann solver of Roe and the solver of Harten--Lax--van Leer (HLL) and its variants, such as the HLLC solver, are widely used as building blocks of finite volume Godunov-type methods for the solution of the Euler equations of gas dynamics and related hyperbolic flow models. The HLLC solver has gained increasing popularity over the last two decades since it possesses some of the good properties of the Roe solver and in addition satisfies important entropy and positivity conditions with no need of special fixes. In this work, we rewrite the classical HLLC Riemann solver in a novel form that highlights the formal mathematical similarity of its wave structure with the one of the Roe solver. This similarity might be useful to extend to the HLLC method some numerical techniques devised specifically for the Roe method. As an example of application, we illustrate the design of a Turkel-type low Mach number preconditioning technique for the HLLC scheme by exploiting methodologies proposed in the literature for the Roe scheme.

Acceleration methods for multi-physics compressible flow

In this work we investigate the Runge–Kutta (RK)/Implicit smoother scheme as a convergence accelerator for complex multi-physics flow problems including turbulent, reactive and also two-phase flows. The flows considered are subsonic, transonic and supersonic flows in complex geometries, and also can be either steady or unsteady flows. All of these problems are considered to be a very stiff.We then introduce an acceleration method for the compressible Navier–Stokes equations. We start with the multigrid method for pure subsonic flow, including reactive flows. We then add the Rossow–Swanson–Turkel RK/Implicit smoother that enables performing all these complex flow simulations with a reasonable CFL number. We next discuss the RK/Implicit smoother for time dependent problem and also for low Mach numbers. The preconditioner includes an intrinsic low Mach number treatment inside the smoother operator. We also develop a modified Roe scheme with a corresponding flux Jacobian matrix. We then give the extension of the method for real gas and reactive flow. Reactive flows are governed by a system of inhomogeneous Navier–Stokes equations with very stiff source terms. The extension of the RK/Implicit smoother requires an approximation of the source term Jacobian. The properties of the Jacobian are very important for the stability of the method. We discuss what the chemical physics theory of chemical kinetics tells about the mathematical properties of the Jacobian matrix. We focus on the implication of the Le-Chatelier's principle on the sign of the diagonal entries of the Jacobian. We present the implementation of the method for turbulent flow. We use a two RANS turbulent model – one equation model – Spalart–Allmaras and a two-equation model – k–ω SST model. The last extension is for two-phase flows with a gas as a main phase and Eulerian representation of a dispersed particles phase (EDP). We present some examples for such flow computations inside a ballistic evaluation rocket motor.The numerical examples in this work include transonic flow about a RAE2822 airfoil, about a M6 Onera wing, NACA0012 airfoil at very low Mach number, two-phase flow inside a Ballistic evaluation motor (BEM), a turbulent reactive shear layer and a time dependent Sod's tube problem.

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax–Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the Hs norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former “shape functions” and “symmetric potential schemes” are highlighted.

A hybrid numerical scheme for aeroheating computation of hypersonic reentry vehicles

Hypersonic aeroheating simulations were conducted for three-dimensional (3D) reentry capsule for the Mars Science Laboratory (MSL) using the AUSM and the Roe schemes. The study confirms that AUSM family of schemes generate unphysical oscillations in the wall heat flux distributions due to the unstable simulation of the shock wave. And the wall heat flux is underpredicted by the Roe scheme due to excess dissipation, even though captures the shock wave accurately. In order to overcome these limitations, we develop a numerical scheme that can capture the shock wave stably and still maintain an accurate prediction of the heat flux. We propose a novel hybrid scheme that combines both the AUSM+ and Roe schemes. Two different hybrid techniques are proposed for switching between the two schemes, one based on the shock wave detection and other utilizing a criterion based on the distance from the wall. The implementation of the two hybrid schemes for the 3D MSL reentry capsule was studied for prediction accuracy compared to experimental data. It is found that the second hybrid scheme (based on the criterion for switching based on distance from the wall) can eliminate the unphysical oscillations in the wall heat flux distributions and improve the aeroheating prediction accuracy.

Cures for expansion shock and shock instability of Roe scheme based on momentum interpolation mechanism

The common defects of the Roe scheme are the non-physical expansion shock and shock instability. By removing the momentum interpolation mechanism (MIM), an improved method with several advantages has been presented to suppress the shock instability. However, it cannot prevent the expansion shock and is incompatible with the traditional curing method for expansion shock. To solve the problem, the traditional curing mechanism is analyzed. Effectiveness of the traditional curing method is discussed, and several defects are identified, one of which leads to incompatibility between curing shock instability and expansion shock. Consequently, an improved Roe scheme is proposed, which is with low computational costs, concise, easy to implement, and robust. More importantly, the proposed scheme can simultaneously solve the problem of shock instability and expansion shock without additional costs.

Numerical Investigation of Flow Around a Multi-Element Airfoil with Hybrid RANS-LES Approaches Based on SST Model

AbstractAccurate prediction of the flow around multi-element airfoil is a prerequisite for improving aerodynamic performance, but its complex flow features impose high demands on turbulence modeling. In this work, delayed detached-eddy-simulation (DDES) and zonal detached-eddy-simulation (ZDES) was applied to simulate the flow past a three-element airfoil. To investigate the effects of numerical dissipation of spatial schemes, the third-order MUSCL and the fifth-order interpolation based on modified Roe scheme were implemented. From the comparisons between the calculations and the available experimental result, third-order MUSCL-Roe can provide satisfactory mean velocity profiles, but the excessive dissipation suppresses the velocity fluctuations level and eliminates the small-scale structures; DDES cannot reproduce the separation near the trailing edge of the flap which lead to the discrepancy in mean pressure coefficients, while ZDES result has better tally with the experiment.

High-resolution simulation and parametric research on helicopter rotor vortex flowfield with TAMI control in hover

In order to investigate the effect of active jet control on the rotor vortex flowfield in hover, a high-accuracy numerical method is presented in this paper, and a novel computer code is developed to calculate rotor flowfield with tip air mass injection. In the present computational fluid dynamics method, the Navier–Stokes equation is selected as the governing equation. For the sake of reducing numerical diffusion in calculation, the upwind Roe scheme with a fifth-order WENO scheme is employed to spatial discretization. A dual-time method is utilized in time marching and the high-efficiency implicit LU-SGS scheme is applied to every pseudo time step. The Baldwin–Lomax turbulence model is employed for eddy viscosity modeling. A surface boundary condition, which may effectively simulate the effect of tip air mass injection, is introduced into the current method. The overset grid method, which can veritably simulate the rotor blade motion, is adopted to exchange the flowfield information between the rotor blade and background grids. By the established method, the rotor flowfield with jet control of different jet angles is simulated. Furthermore, parametric analyses of jet velocity are carried out at different tip March numbers and collective pitches. It is demonstrated that tip air mass injection can make the tip vortex rolled up and generated in advance compared with the clean rotor, which can change the position and strength of the vortex core greatly. It is also shown that the rotor aerodynamic characteristics, such as blade surface pressure and radial lift distributions, are more sensitive to tip air mass injection control at the condition with a lower Mach number or a lower collective pitch.

A $$\varvec{2\times 2}$$ hyperbolic system modelling incompressible two phase flows: theory and numerics

We propose a \(2\times 2\) hyperbolic system of conservation laws to model the dynamics of two incompressible fluids in mechanical disequilibrium. In the theoretical part of the paper we show that this 1D system is not strictly hyperbolic, that the characteristic speed can not a priori be ordered and that the characteristic fields are neither genuinely nonlinear, nor linearly degenerate. We nevertheless prove the existence and uniqueness of an admissible solution to the Riemann problem. This solution remains bounded with positive volume fractions even when one the phases vanishes. We conclude that the multiphase/single phase transition does not imply mechanical equilibrium but displays a non classical wave structure. In the numerical part of the paper we propose some approximate Riemann solvers to simulate the model, especially the multiphase/single phase transition. The classical Riemann solvers have been considered as Godunov scheme, Roe scheme with or without entropy fix. We also propose an in-cell discontinuous reconstruction method which proves to be successful, whereas the other schemes may show some spurious oscillations in some Riemann problem. Finally, as an application we study and simulate the problem of phase separation by gravity.

Implementation of dual time-stepping strategy of the gas-kinetic scheme for unsteady flow simulations.

In our study, the dual time-stepping strategy of the gas-kinetic scheme is constructed and used for the simulation of unsteady flows. In comparison to the previous implicit gas-kinetic scheme, both the inviscid and viscous flux Jacobian are considered in our work, and the linear system of the pseudo-steady-state is solved by applying generalized minimal residual algorithm. The accuracy is validated by several numerical cases, the incompressible flow around blunt bodies (stationary circular cylinder and square cylinder), and the transonic buffet on the NACA0012 airfoil under hybrid mesh. The numerical cases also demonstrate that the present method is applicable to approach the fluid flows from laminar to turbulent and from incompressible to compressible. Finally, the case of acoustic pressure pulse is carried out to evaluate the effects of enlarged time step, and the side effect of enlarged time step is explained. Compared with the explicit gas-kinetic scheme, the proposed scheme can greatly accelerate the computation and reduce the computational costs for unsteady flow simulations.

An implicit turbulence model for low-Mach Roe scheme using truncated Navier–Stokes equations

The original Roe scheme is well-known to be unsuitable in simulations of turbulence because the dissipation that develops is unsatisfactory. Simulations of turbulent channel flow for Reτ=180 show that, with the ‘low-Mach-fix for Roe’ (LMRoe) proposed by Rieper [J. Comput. Phys. 230 (2011) 5263–5287], the Roe dissipation term potentially equates the simulation to an implicit large eddy simulation (ILES) at low Mach number. Thus inspired, a new implicit turbulence model for low Mach numbers is proposed that controls the Roe dissipation term appropriately. Referred to as the automatic dissipation adjustment (ADA) model, the method of solution follows procedures developed previously for the truncated Navier–Stokes (TNS) equations and, without tuning of parameters, uses the energy ratio as a criterion to automatically adjust the upwind dissipation. Turbulent channel flow at two different Reynold numbers and the Taylor–Green vortex were performed to validate the ADA model. In simulations of turbulent channel flow for Reτ=180 at Mach number of 0.05 using the ADA model, the mean velocity and turbulence intensities are in excellent agreement with DNS results. With Reτ=950 at Mach number of 0.1, the result is also consistent with DNS results, indicating that the ADA model is also reliable at higher Reynolds numbers. In simulations of the Taylor–Green vortex at Re=3000, the kinetic energy is consistent with the power law of decaying turbulence with −1.2 exponents for both LMRoe with and without the ADA model. However, with the ADA model, the dissipation rate can be significantly improved near the dissipation peak region and the peak duration can be also more accurately captured. With a firm basis in TNS theory, applicability at higher Reynolds number, and ease in implementation as no extra terms are needed, the ADA model offers to become a promising tool for turbulence modeling.

Effects of non-equilibrium condensation on aerodynamics of the flow field in a steam turbine cascade

In the present paper an in-house CFD code is developed using Roe scheme to simulate condensing two-phase flow in blade to blade passage of a steam turbine. Effects of condensation on the flow field of steam turbine rotor tip section are investigated for different outlet pressures. Firstly, comparison is performed between results of wet and dry cases. Then effects of outlet pressure variations on the flow field are studied. Finally effects of condensation on different specifications of the flow field (total pressure loss coefficient, entropy generation and deviation angle) are investigated. Also the mechanism of flow deviation in the cascade flow field is described. Condensation has a great influence on the behavior of the flow field based on the numerical results of this paper. It changes the outflow direction and consequently the flow entering to the next blade deviates from its on-design condition, thus additional losses are produced. For example, the value of deviation angle reaches to 7.62for wet case and exit Mach number Me=1.45. Also there are stagnation pressure loss and entropy generation due to non-equilibrium condensation that reduce the efficiency of the steam turbine

Investigations of a compressible second order finite volume code towards the incompressible limit

The accuracy of many existing codes, originally implemented for transonic or supersonic problems, often degrades as the inflow Mach number tends to zero. At low Mach number, the Roe scheme or a similar formulated artificial viscosity scheme which is close to a Roe scheme presents an excess of artificial viscosity. As a result, several years ago a correction of these kinds of schemes was introduced by Turkel. Such correction yields a significant improvement of the accuracy of the solution for the low Mach regime. In particular, for inviscid low Mach number flows it was shown in the literature that the accuracy of the results obtained with the modification of Turkel is comparable with results obtained with CFD codes designed for incompressible flows. Unfortunately, a loss of robustness for the correction of Turkel is often observed when considering compressible high Reynolds number viscous flows. Typical examples investigated in aerodynamics are configurations at high angle of attack, for example an aircraft in landing configuration. Besides other complex flow phenomena, for these kinds of flows both strong locally compressible flows and large incompressible regions are observed. In particular for these kinds of flows the modification of the code needs to work reliably. Based on the work of Rossow and Swanson we are going to present a further modification for the extension of a compressible code towards the incompressible limit. Numerical examples demonstrate the superiority of the method when compared with the modification suggested by Turkel. Moreover, 2D and 3D examples are given demonstrating the applicability of the method to flows of locally compressible and large incompressible effects. Furthermore, the suggested method does not show severe robustness problems for these kinds of flows when compared with the non-modified scheme.