Dynamic stability of the high-head Francis turbines is one of the challenging problems. Unsteady rotor–stator interaction (RSI) develops dynamic stresses and leads to crack in the blades. In a high-head turbine, vaneless space is small and the amplitudes of RSI frequencies are very high. Credible estimation of the amplitudes is vital for the runner design. The current study is aimed to investigate the amplitudes of RSI frequencies considering a compressible flow. The hydro-acoustic phenomenon is dominating the turbines, and the compressibility effect should be accounted for accurate estimation of the pressure amplitudes. Unsteady pressure measurements were performed in the turbine during the best efficiency point (BEP) and part load (PL) operations. The pressure data were used to validate the numerical model. The compressible flow simulations showed 0.5–3% improvement in the time-averaged pressure and the amplitudes over incompressible flow. The maximum numerical errors in the vaneless space and runner were 6% and 10%, respectively. Numerical errors in the instantaneous pressure amplitudes at the vaneless space, runner, and draft tube were ±1.6%, ±0.9%, and ±1.8%, respectively. In the draft tube, the incompressible flow study showed the pressure amplitudes up to eight times smaller than those of the compressible. Unexpectedly, the strong effect of RSI was seen in the upper and lower labyrinth seals, which was absent for the incompressible flow.

Production from a high-pressure gas well at a high production rate encounters the risk of wellbore tensile failure when the pressure gradient of the averaged gas flow becomes large. At the pore-scale, however, when flow in just one pore is choked, gas pressure gradient at the point of choking becomes singular, leading to an unbounded average of the pressure gradient. This study investigates the choking condition for compressible gas flow in a single pore. It is found that wellbore tensile failure can occur at a much lower inlet-to-outlet pressure ratio than predicted from the macroscopic theory of porous medium flow.

This paper presents a hybrid compressible–incompressible approach for simulating the Richtmyer–Meshkov instability (RMI) and associated mixing. The proposed numerical approach aims to circumvent the numerical deficiencies of compressible methods at low Mach (LM) numbers, when the flow has become essentially incompressible. A compressible flow solver is used at the initial stage of the interaction of the shock wave with the fluids interface and the development of the RMI. When the flow becomes sufficiently incompressible, based on a Mach number prescribed threshold, the simulation is carried out using an incompressible flow solver. Both the compressible and incompressible solvers use Godunov-type methods and high-resolution numerical reconstruction schemes for computing the fluxes at the cell interfaces. The accuracy of the model is assessed by using results for a two-dimensional (2D) single-mode RMI.

Slip flow in various noncircular microchannels has been further examined, and a simple model for a normalized Poiseuille number is proposed. As for slip flow, no solutions or graphical and tabulated data exist for most geometries; the developed simple model fills this void and can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution, and pressure drop of slip flow in noncircular microchannels by the research community for the practical engineering design of microchannels. The incompressible flow criterion for gas flow in microchannels is given. A Mach number less than 0.3 is not sufficient to ensure that the flow is incompressible. Compressibility depends on the product of two dimensionless parameters: L / L ( D Re ) ( D Re ) and Ma (Arkilic et al., 1997, “Gaseous Slip Flow in Long Microchannels,” J. Microelectromech. Syst., 6 (2), pp. 167–178). Some flows where Ma < 0.3 are low speed compressible flows. A fresh general pressure drop model for isothermal low Mach number compressible flow in microchannels is proposed. If the pressure drop is less than 10% of the outlet pressure, the flow can be considered as incompressible for practical engineering applications. This paper improves and extends previous studies on slip flow in noncircular microchannels.

In general, the computation of single phase subsonic mass velocity of gas flowing through a pipe requires a computerized iterative analysis. The equations for the friction factor for laminar and turbulent flow are used to obtain explicit equations for the subsonic mass velocity as a function of the pressures at the ends of a pipe. Explicit equations for mass velocity are presented. Included within the equations is a heat transfer ratio, which can vary between 0 for adiabatic flow conditions to 1 for isothermal flow conditions. The use of this heat transfer ratio also enables the formulation of an explicit equation for the gas temperature along the pipe for nonisothermal flow conditions. The explicit equations eliminate the need for an iterative solution. Laboratory data are used to support the accuracy of the model.

For the aim of computing compressible turbulent flowfield involving shock waves, an implicit large eddy simulation (LES) code has been developed based on the idea of monotonically integrated LES. We employ the weighted compact nonlinear scheme (WCNS) not only for capturing possible shock waves but also for attaining highly accurate resolution required for implicit LES. In order to show that WCNS is a proper choice for implicit LES, a two-dimensional homogeneous turbulence is first obtained by solving the Navier–Stokes equations for incompressible flow. We compare the inertial range in the computed energy spectrum with that obtained by the direct numerical simulation (DNS) and also those given by the different LES approaches. We then obtain the same homogeneous turbulence by solving the equations for compressible flow. It is shown that the present implicit LES can reproduce the inertial range in the energy spectrum given by DNS fairly well. A truncation of energy spectrum occurs naturally at high wavenumber limit indicating that dissipative effect is included properly in the present approach. A linear stability analysis for WCNS indicates that the third order interpolation determined in the upwind stencil introduces a large amount of numerical viscosity to stabilize the scheme, but the same interpolation makes the scheme weakly unstable for waves satisfying k Δ x ≈ 1 . This weak instability results in a slight increase in the energy spectrum at high wavenumber limit. In the computed result of homogeneous turbulence, a fair correlation is shown to exist between the locations where the magnitude of ∇ × ω becomes large and where the weighted combination of the third order interpolations in WCNS deviates from the optimum ratio to increase the amount of numerical viscosity. Therefore, the numerical viscosity involved in WCNS becomes large only at the locations where the subgrid-scale viscosity can arise in ordinary LES. This suggests the reason why the present implicit LES code using WCNS can resolve turbulent flowfield reasonably well.

Recent progress in understanding the theoretical basis and effectiveness of implicit large eddy simulation (ILES) is reviewed in both incompressible and compressible flow regimes. We use a modified equation analysis to show that the leading-order truncation error terms introduced by certain hybrid high resolution methods provide implicit subgrid scale (SGS) models similar in form to those of conventional mixed SGS models. Major properties of the implicit SGS model are related to the choice of high-order and low-order scheme components, the choice of a flux limiter, which determines how these schemes are blended locally depending on the flow, and the designed balance of the dissipation and dispersion contributions to the numerical solution. Comparative tests of ILES and classical LES in the Taylor–Green vortex case show robustness in capturing established theoretical findings for transition and turbulence decay.

The present study concerns the application of large eddy simulation (LES) and implicit LES (ILES) to engineering flow problems. Such applications are often very complicated, involving both complex geometries and complex physics, such as turbulence, chemical reactions, phase changes, and compressibility. The aim of the study is to illustrate what problems occur when attempting to perform such engineering flow calculations using LES and ILES, and put these in relation to the issues originally motivating the calculations. The issues of subgrid modeling are discussed with particular emphasis on the complex physics that needs to be incorporated into the LES models. Results from representative calculations, involving incompressible flows around complex geometries, aerodynamic noise, compressible flows, combustion, and cavitation, are presented, discussed, and compared with experimental data whenever possible.

Implicit large eddy simulation (ILES) has provided many computer simulations with an efficient and effective model for turbulence. The capacity for ILES has been shown to arise from a broad class of numerical methods with specific properties producing nonoscillatory solutions using limiters that provide these methods with nonlinear stability. The use of modified equation has allowed us to understand the mechanisms behind the efficacy of ILES as a model. Much of the understanding of the ILES modeling has proceeded in the realm of incompressible flows. Here, we extend this analysis to compressible flows. While the general conclusions are consistent with our previous findings, the compressible case has several important distinctions. Like the incompressible analysis, the ILES of compressible flow is dominated by an effective self-similarity model ( Bardina, J., Ferziger, J. H., and Reynolds, W. C., 1980, “Improved Subgrid Scale Models for Large Eddy Simulations,” AIAA Paper No. 80–1357 ; Borue, V., and Orszag, S. A., 1998, “Local Energy Flux and Subgrid-Scale Statistics in Three Dimensional Turbulence,” J. Fluid Mech., 366, pp. 1–31 ; Meneveau, C., and Katz, J., 2000, “Scale-Invariance and Turbulence Models for Large-Eddy Simulations,” Annu. Rev. Fluid. Mech., 32, pp. 1–32 ). Here, we focus on one of these issues, the form of the effective subgrid model for the conservation of mass equations. In the mass equation, the leading order model is a self-similarity model acting on the joint gradients of density and velocity. The dissipative ILES model results from the limiter and upwind differencing resulting in effects proportional to the acoustic modes in the flow as well as the convective effects. We examine the model in several limits including the incompressible limit. This equation differs from the standard form found in the classical Navier–Stokes equations, but generally follows the form suggested by Brenner (2005, “Navier-Stokes Revisited,” Physica A, 349(1–2), pp. 60–133) in a modification of Navier–Stokes necessary to successfully reproduce some experimentally measured phenomena. The implications of these developments are discussed in relation to the usual turbulence modeling approaches.

A commercial computational fluid dynamics (CFD) package was used to develop a three-dimensional, fully turbulent model of the compressible flow across a complex-geometry venturi, such as those typically found in small engine carburetors. The results of the CFD simulations were used to understand the effect of the different obstacles in the flow on the overall discharge coefficient and the static pressure at the tip of the fuel tube. It was found that the obstacles located at the converging nozzle of the venturi do not cause significant pressure losses, while those obstacles that create wakes in the flow, such as the fuel tube and throttle plate, are responsible for most of the pressure losses. This result indicated that an overall discharge coefficient can be used to correct the mass flow rate, while a localized correction factor can be determined from three-dimensional CFD simulations in order to estimate the static pressure at locations of interest within complex venturis.

The ability to accurately predict the aerodynamic torque and lift and drag forces on a 2-D model of a 0.18 aspect ratio biconvex circular-arc disk operating in a compressible flow using computational fluid dynamics (CFD) was investigated. Fluent 6.0 was the CFD package utilized to perform these calculations. Grid-convergence and time-convergence/stability were analyzed first, followed by a qualitative study of the Spalart-Allmaras, k - ε , and k - ω turbulence models with their enhancement features and model variants. Fluent was used to predict the pressure profile on the disk surface for disk positions 30, 45, and 60 deg (where 0 deg is the fully closed position) and over a range of pressure ratios. The pressure ratios were selected to determine the capability of CFD to accurately predict the flow field and resulting torque in flows ranging from nearly incompressible to highly compressible. Fluent predictions for the pressure profiles on the disk were compared to test data so that the lift and drag forces and aerodynamic torque could be determined responsibly. Acceptable comparisons were noted.

Steady and unsteady shock boundary-layer interactions are studied numerically by solving the two-dimensional time-dependent Navier-Stokes equations. To validate the numerical method, the steady interaction is compared with measurements and other numerical results reported in the literature. The numerical study of the steady interaction leads to a suitable method for transpiration boundary conditions. The method applies to unsteady flows as well. Using the validated numerical method, we show that an unsteady shock boundary-layer interaction can occur in a supersonic flow over a flat plate subjected to suction and blowing from the opposite side of the plate, even though the imposed transpiration is steady. Depending on the Mach number, the Reynolds number, the distance of the transpiration boundary to the lower wall, and the transpiration profile, the unsteadiness can be inviscid or viscous dominated. The viscous effect is characterized by the occurrence of self-excited vortex shedding. A criterion for the onset of vortex shedding for internal compressible flows is also proposed.

In this contribution we consider large-eddy simulation (LES) using the high-pass filtered (HPF) Smagorinsky model of a spatially developing supersonic turbulent boundary layer at a Mach number of 2.5 and momentum-thickness Reynolds numbers at inflow of ∼ 4500 . The HPF eddy-viscosity models employ high-pass filtered quantities instead of the full velocity field for the computation of the subgrid-scale (SGS) model terms. This approach has been proposed independently by Vreman ( Vreman, A. W., 2003, Phys. Fluids, 15, pp. L61–L64 ) and Stolz et al. (Stolz, S., Schlatter, P., Meyer, D., and Kleiser, L., 2003, in Direct and Large Eddy Simulation V, Kluwer, Dordrecht, pp. 81–88 ). Different from classical eddy-viscosity models, such as the Smagorinsky model ( Smagorinsky, J., 1963, Mon. Weath. Rev, 93, pp. 99–164 ) or the structure-function model ( Métais, O. and Lesieur, M., 1992, J. Fluid Mech., 239, pp. 157–194 ) which are among the most often employed SGS models for LES, the HPF eddy-viscosity models do need neither van Driest wall damping functions for a correct prediction of the viscous sublayer of wall-bounded turbulent flows nor a dynamic determination of the coefficient. Furthermore, the HPF eddy-viscosity models are formulated locally and three-dimensionally in space. For compressible flows the model is supplemented by a HPF eddy-diffusivity ansatz for the SGS heat flux in the energy equation. Turbulent inflow conditions are generated by a rescaling and recycling technique in which the mean and fluctuating part of the turbulent boundary layer at some distance downstream of inflow is rescaled and reintroduced at the inflow position ( Stolz, S. and Adams, N. A., 2003, Phys. Fluids, 15, pp. 2389–2412 ).

Recently, Eulerian methods for capturing interfaces in multi-fluid problems become increasingly popular. While these methods can effectively handle significant deformations of interface, the treatment of the boundary conditions in certain classes of compressible flows are known to produce nonphysical oscillations due to the radical change in equation of state across the material interface. One promising recent development to overcome these problems is the Ghost Fluid Method (GFM). The present study initiates a new methodology for boundary condition capturing in multifluid compressible flows. The method, named Characteristics-Based Matching (CBM), capitalizes on recent developments of the level set method and related techniques, i.e., PDE-based re-initialization and extrapolation, and the Ghost Fluid Method (GFM). Specifically, the CBM utilizes the level set function to capture interface position and a GFM-like strategy to tag computational nodes. In difference to the GFM method, which employs a boundary condition capturing in primitive variables, the CBM method implements boundary conditions based on a characteristic decomposition in the direction normal to the boundary. In this way overspecification of boundary conditions is avoided and we believe so will be spurious oscillations. In this paper, we treat (moving or stationary) fluid-solid interfaces and present numerical results for a select set of test cases. Extension to fluid-fluid interfaces will be presented in a subsequent paper.

The non-ideal gas equations and mathematical formulation developed in this work using the compressibility factor in the equation of state closely resemble the derivations used for the ideal gas mathematical formulation for a direct comparison of the differences between the ideal versus the non-ideal gas law. The local Mach number is defined for the non-ideal gas. The plenum total variables used in compressible flow are expressed in terms of the local Mach number for the polytrope and Rayleigh models. A power law relationship is derived between the thermodynamic variables that allow an analytical result for the mass flow under certain constraints.

The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number ( Pr ) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively.

The approximate deconvolution model for large-eddy simulation is formulated for a second-order finite volume scheme. With the approximate deconvolution model, an approximation of the unfiltered solution is obtained by repeated filtering, and given a good approximation of the unfiltered solution, the nonlinear terms of the Navier-Stokes equations are computed directly. The effect of scales not represented on the numerical grid is modeled by a relaxation regularization involving a secondary filter operation. A turbulent channel flow at a Mach number of M = 1.5 and a Reynolds number based on bulk quantities of Re=3000 is selected for validation of the approximate deconvolution model implementation in a finite volume code. A direct numerical simulation of this configuration has been computed by Coleman et al. Overall, our large-eddy simulation results show good agreement with our filtered direct numerical simulation data. For this rather simple configuration and the low-order spatial discretization, differences between approximate deconvolution model and a no-model computation are found to be small.