Low Mach Number Preconditioning Research Articles

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.

A Robust Mixing Plane and Its Application in Three-Dimensional Inverse Design of Transonic Turbine Stages

A robust mixing plane method satisfying interface flux conservation, nonreflectivity and retaining interface flow variation; valid at all Mach numbers and applicable for any machine configuration is formulated and implemented in a vertex based finite volume solver for flow analysis and inverse design. The formulation is based on superposing perturbed flow variables derived from the three-dimensional (3D) characteristics obtained along the flow direction on the exchanged mixed out averaged quantities. The method is extended for low speed applications using low Mach number preconditioning. Subsequently, inverse design runs over a single stage transonic low pressure (LP) turbine configuration conducted at a fixed mass flow boundary condition and spanwise loading condition similar to the baseline generates optimized configurations providing performance improvement while achieving prespecified target meridional load distribution.

A Computational Study on the Aerodynamic Influence of a Propeller on an MAV by Unstructured Overset Grid Technique and Low Mach Number Preconditioning

The influence of a propeller on the aerodynamic performance of an MAV is investigated using an unstructured overset grid technique. The flow regime of a fixed-wing MAV powered by a propeller contains both incompressible regions due to the low flight speed, as well as compressible flow areas near the propeller-tip region. In order to simulate all speed flow efficiently, a dual-time preconditioning method is employed in the present study. The methodology in this paper is verified as providing a reliable numerical simulation tool for all flow regimes, in the additional presence of moving boundaries, which is treated with an overset grid approach.

Numerical wave breaking with macro-roughness

This paper reports on a numerical investigation of solitary wave breaking over a sloping bottom covered with macro-roughness elements. Wave breaking is simulated by solving Euler equations with a two phase incompressible flow model. The hyperbolic system of the conservation laws is solved with a finite volume discretization on an unstructured grid. An artificial compressibility approach allows the use of a fully explicit scheme for an efficient parallel implementation. The numerical model is based on a low Mach number preconditioning and a second order Riemann solver. Several test cases are performed to analyze the role played by macro-roughness on the breaking dynamics. The influence of the macro-roughness elements on the solitary long wave breaking is shown to depend on two dimensionless ratios D/A and H/A, where D, H and A are the separation distances between the macro-roughness elements, the height of the elements and the wave amplitude, respectively. Significant effects are observed for large values of D/A and H/A. The successive cycles of impact/splash-ups/rebounds are strongly impaired. Three-dimensional wave breaking simulations are presented, showing the robustness of the method for modeling complex wave-structure interactions.

A Comparative Investigation of Round and Fan-Shaped Cooling Hole Near Flow Fields

This study presents velocity and turbulence data measured experimentally in the near field of a round and a laterally expanded fan-shaped cooling hole. Both holes are fed by a plenum inlet, and interact with a turbulent mainstream boundary layer. Flow is Reynolds number matched to engine conditions to preserve flow structure, and two coolant to mainstream blowing momentum ratios are investigated experimentally. Results clearly identify regions of high shear for the round hole as the jet penetrates into the mainstream. In contrast, the distinct lack of high shear regions for the fan-shaped hole points to reasons for improvements in cooling performance noted by previous studies. Two different computational fluid dynamics codes are used to predict the flow within and downstream of the fan-shaped hole, with validation from the experimental measurements. One code is the commercially available ANSYS CFX 10.0, and the other is the density-based solver with low Mach number preconditioning, HYDRA, developed in-house by Rolls-Royce plc for high speed turbomachinery flows. Good agreement between numerical and experimental data for the center-line traverses was obtained for a steady state solution, and a region of reversed flow within the expansion region of the fan-shaped hole was identified.

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

AbstractDiscrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.

Large-eddy simulation of film cooling flows at density gradients

The present paper investigates the impact of the velocity and density ratio on the turbulent mixing process in gas turbine blade film cooling. A cooling fluid is injected through an inclined pipe at α = 30° into a turbulent boundary layer at a freestream Reynolds number of Re ∞ = 400,000. This jet-in-a-crossflow (JICF) problem is investigated using large-eddy simulations (LES). The governing equations comprise the Navier–Stokes equations plus additional transport equations for several species to simulate a non-reacting gas mixture. That is, gases of different density are effused into an air crossflow at a constant temperature. An efficient large-eddy simulation method for low subsonic flows based on an implicit dual time-stepping scheme combined with low Mach number preconditioning is applied. The comparison of the numerical findings with experimental velocity data from two-component particle-image velocimetry (PIV) measurements shows an excellent agreement. The results evidence the dynamics of the flow field in the vicinity of the jet hole, i.e., the recirculation region and the inclination of the shear layers, to be mainly determined by the velocity ratio. However, evaluating the cooling efficiency downstream of the jet hole the mass flux ratio proves to be the dominant similarity parameter, i.e., the density ratio of the jet and crossflow fluid has to be considered.

Large-eddy Simulation of Film Cooling Flows with Variable Density Jets

The present paper investigates the impact of the velocity and density ratio on the turbulent mixing process in gas turbine blade film cooling. A cooling fluid is injected from an inclined pipe at α=30° into a turbulent boundary layer profile at a freestream Reynolds number of Re ∞ = 400,000. This jet-in-a-crossflow (JICF) problem is investigated using large-eddy simulations (LES). The governing equations comprise the Navier–Stokes equations plus additional transport equations for several species to simulate a non-reacting gas mixture. A variation of the density ratio is simulated by the heat-mass transfer analogy, i.e., gases of different density are effused into an air crossflow at a constant temperature. An efficient large-eddy simulation method for low subsonic flows based on an implicit dual time-stepping scheme combined with low Mach number preconditioning is applied. The numerical results and experimental velocity data measured using two-component particle-image velocimetry (PIV) are in excellent agreement. The results show the dynamics of the flow field in the vicinity of the jet hole, i.e., the recirculation region and the inclination of the shear layers, to be mainly determined by the velocity ratio. However, evaluating the cooling efficiency downstream of the jet hole the mass flux ratio proves to be the dominant similarity parameter, i.e., the density ratio between the fluids and the velocity ratio have to be considered.

A preconditioned dual time-stepping procedure coupled with matrix-free LU-SGS scheme for unsteady low speed viscous flows with moving objects

In this paper an effective method is developed to solve unsteady low speed viscous flow problems with moving objects by using the governing equations of compressible fluids. The method is based on a dual time-stepping scheme, combined with low Mach number preconditioning and an implicit matrix-free Lower-Upper Symmetric Gauss-Seidel iteration on unstructured dynamic meshes. Because preconditioning modifies the governing equations, that induces the change of system's eigenvalues and eigenvectors, characteristic boundary conditions are also modified to suit the preconditioned characteristic system. Several test cases are simulated, including an in-line oscillating cylinder in a fluid at rest, flow over a flapping NACA0014 airfoil and low speed flow past a flapping-wing micro-air vehicle. Compared with experimental results whenever possible, the computed results indicate that this algorithm shows satisfactory improvement of solution efficiency and accuracy for low speed flow problems.

Efficient computation of compressible and incompressible flows

The combination of explicit Runge–Kutta time integration with the solution of an implicit system of equations, which in earlier work demonstrated increased efficiency in computing compressible flow on highly stretched meshes, is extended toward conditions where the free stream Mach number approaches zero. Expressing the inviscid flux Jacobians in terms of Mach number, an artificial speed of sound as in low Mach number preconditioning is introduced into the Jacobians, leading to a consistent formulation of the implicit and explicit parts of the discrete equations. Besides extension to low Mach number flows, the augmented Runge–Kutta/Implicit method allowed the admissible Courant–Friedrichs–Lewy number to be increased from O(1 0 0) to O(1 0 0 0). The implicit step introduced into the Runge–Kutta framework acts as a preconditioner which now addresses both, the stiffness in the discrete equations associated with highly stretched meshes, and the stiffness in the analytical equations associated with the disparity in the eigenvalues of the inviscid flux Jacobians. Integrated into a multigrid algorithm, the method is applied to efficiently compute different cases of inviscid flow around airfoils at various Mach numbers, and viscous turbulent airfoil flow with varying Mach and Reynolds number. Compared to well tuned conventional methods, computation times are reduced by half an order of magnitude.

On Low Mach Number Preconditioning of Finite Volume Schemes

AbstractA finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments, as well as by analysis, that the preconditioned scheme yields a physically corrected pressure distribution and combined with an explicit time integrator it is stable if the time step Δt satisfies the requirement to be 𝒪(M 2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Δt = 𝒪(M ),M → 0, though producing unphysical results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A large-eddy simulation method for low Mach number flows using preconditioning and multigrid

Large-eddy simulation (LES) has become one of the major tools to investigate the physics of turbulent compressible and incompressible flows. At low Mach numbers the performance of LES codes developed for the conservation equations of compressible fluids deteriorates due to the presence of two different time scales associated with acoustic and convective waves. In many subsonic turbulent flows low Mach number regions exist, which require large integration times until a fully developed flow is established. In such cases, the efficiency of algorithms for compressible flows can be improved considerably by low Mach number preconditioning methods. In this paper an efficient method of solution for low subsonic flows is developed based on an implicit dual time stepping scheme combined with low Mach number preconditioning and a multigrid acceleration technique. To validate the efficiency and the accuracy of the method, large-eddy simulations of turbulent channel flow at Re τ = 590 and cylinder flow at Re = 3900 are performed for several Mach numbers and the data are compared with numerical and experimental findings from the literature. The speedup compared to a purely explicit approach is in the range of 6–40.

Large eddy simulation of heated vertical annular pipe flow in fully developed turbulent mixed convection

The goal of the present study was to perform a large eddy simulation of vertical turbulent annular pipe flow under conditions in which fluid properties vary significantly, and to investigate the effects of buoyancy on the turbulent structures and transport. Isoflux wall boundary conditions with low and high heating are imposed. The compressible filtered Navier–Stokes equations are solved using a second order accurate finite volume method. Low Mach number preconditioning is used to enable the compressible code to work efficiently at low Mach numbers. A dynamic subgrid-scale stress model accounts for the subgrid-scale turbulence. Comparisons were made with available experimental data. The results showed that the strong heating and buoyant force caused distortions of the flow structure resulting in reduction of turbulent intensities, shear stress, and turbulent heat flux, particularly near the wall.

Convergence acceleration for computing steady-state compressible flow at low Mach numbers

A novel technique is introduced to accelerate the convergence of compressible low Mach number flow computations to the steady state. The stiffness due to the large disparity of flow velocity and acoustic wave speeds is bypassed by artificially reducing the speed of sound and thereby increasing the Mach number. This Mach number transformation is achieved by subtracting a constant value from the pressure in the entire flow field. Only the inviscid terms of the energy equation are influenced by that pressure decrease. The steady-state error is corrected by solving a scalar equation after each time step such that the steady-state solutions of the modified and non-modified schemes coincide. Thus, for each low Mach number simulation, one can obtain a convergence performance comparable to the corresponding simulation with a Mach number of about 0.4. This convergence acceleration is demonstrated for premixed laminar flames. If the present technique is implemented without time splitting, it corresponds to a novel low Mach number preconditioning.