Man Mohan Rai

A conservative treatment of zonal boundaries for Euler equation calculations

Finite-difference calculations require the generation of a grid for the region of interest. A zonal approach, wherein the given region is subdivided into zones and the grid for each zone is generated independently, makes the grid-generation process for complicated topologies and for regions requiring selective grid refinement a fairly simple task. This approach results in new boundaries within the given region, that is, zonal boundaries at the interfaces of the various zones. The zonal-boundary scheme (the integration scheme used to update the points on the zonal boundary) for the Euler equations must be conservative, accurate, stable, and applicable to general curvilinear coordinate systems. A zonal-boundary scheme with these desirable properties is developed in this study. The scheme is designed for explicit, first-order-accurate integration schemes but can be modified to accommodate second-order-accurate explicit and implicit integration schemes. Results for inviscid flow, including supersonic flow over a cylinder, blast-wave diffraction by a ramp, and one-dimensional shock-tube flow are obtained on zonal grids. The conservative nature of the zonal-boundary scheme permits the smooth transition of the discontinuities associated with these flows from one zone to another. The calculations also demonstrate the continuity of contour lines across zonal boundaries that can be achieved with the present zonal scheme.

Navier-Stokes Simulations of Blade-Vortex Interaction Using High-Order-Accurate Upwind Schemes

Conventional second-order-accurate finite-difference schemes are much too dissipative for computations involving vortices that travel large distances (relative to some measure of the size of the vortex). This study uses a fifth-order-accurate upwind-biased scheme that preserves vortex structure much longer than second-order-accurate, central or upwind difference schemes. Vortex computations demonstrating this aspect of the fifth-order scheme are presented. The method is then applied to a blade-vortex interaction problem. The computed results are in good agreement with earlier numerical computations that required a priori information regarding vortex structure and position throughout the interaction period in order to sustain the vortex. High-order-accurate methods such as the one used in this study may prove essential to computing both nearfield aerodynamics and farfield acoustics associated with blade-vortex interactions as well as other acoustical phenomena.

A relaxation approach to patched-grid calculations with the Euler equations

Abstract A conservative zonal-boundary condition that was used with explicit integration schemes is extended to implicit, upwind, relaxation schemes; in particular to the Osher scheme. The rate of convergence was found to increase considerably with the use of the implicit, relaxationzonal-scheme when compared to the explicit scheme. The relaxation-zonal scheme has also been used in a time-accurate mode. Results demonstrating the time accuracy of the scheme and the feasibility of performing calculations in cases where some parts of the given system move relative to others (e.g., rotor-stator configurations) are presented.

An implicit, conservative, zonal-boundary scheme for Euler equation calculations

Abstract A “zonal”, or “patched-grid”, approach is one in which the flow region of interest is divided into subregions which are then discretized independently, using existing grid generators. The equations of motion are integrated in each subregion in conjunction with zonal-boundary schemes which allow proper information transfer across interfaces that separate subregions. The zonal approach greatly simplifies the treatment of complex geometries and also the addition of grid points to selected regions of the flow. In this study a conservative, zonal-boundary condition that could be used with explicit schemes has been extended so that it can be used with existing second-order accurate implicit integration schemes such as the Beam-Warming and Osher schemes. In the test case considered, the implicit schemes increased the rate of convergence considerably (by a factor of about 30 over that of the explicit scheme). Results demonstratiting the time-accuracy of the zonal scheme and the feasibility of performing calculations on zones that move relative to each other are also presented.

Application of Artificial Neural Networks to the Design of Turbomachinery Airfoils

The feasibility of applying artie cial neural networks to the aerodynamic design of turbomachinery airfoils is investigated. The design process involves dee ning a target pressure distribution, computing several e ows to adequately populate the design space in the vicinity of the target, training the neural network with this data, and, e nding a design that has a pressure distribution that is closest to the target. The last step is carried out using the network as a function evaluator. This design process is tested using an established e ow simulation procedure, a simple two-layer feedforward network and a conjugate gradient optimization technique. Results are presented for some validation tests as well as a complete design effort where the pressure distribution from a modern Pratt and Whitneyturbinewasused as a target. Theseresultsareveryencouraging and clearly warrant furtherdevelopment of the process for full three-dimensional design.

A computational investigation of the instability of the detached shear layers in the wake of a circular cylinder

Cylinder wakes have been studied extensively over several decades to better understand the basic flow phenomena encountered in such flows. The physics of the very near wake of the cylinder is perhaps the most challenging of them all. This region comprises the two detached shear layers, the recirculation region and wake flow. A study of the instability of the detached shear layers is important because these shear layers have a considerable impact on the dynamics of the very near wake. It has been observed experimentally that during certain periods of time that are randomly distributed, the measured fluctuating velocity component near the shear layers shows considerable amplification and it subsequently returns to its normal level (intermittency). Here, direct numerical simulations are used to accomplish a number of objectives such as confirming the presence of intermittency (computationally) and shedding light on processes that contribute significantly to intermittency and shear-layer transition/breakdown. Velocity time traces together with corresponding instantaneous vorticity contours are used in deciphering the fundamental processes underlying intermittency and shear-layer transition. The computed velocity spectra at three locations along the shear layer are provided. The computed shear-layer frequency agrees well with a power-law fit to experimental data.

TWO-DIMENSIONAL COMPUTATIONS OF MULTI-STAGE COMPRESSOR FLOWS USING A ZONAL APPROACH

A clear understanding of the fluid dynamics associated with rotor/stator configurations can be very helpful when optimizing the performance of turbomachinery. In this study, a two-dimensional, implicit, thin-layer, Navier-Stokes zonal approach has been used to investigate the flow within a 2 1/2-stage compressor. Relative motion between the rotor and stator airfoils is made possible with the use of systems of patched and overlaid grids that move with respect to each other. The treatment of multistage turbomachines with arbitrary numbers of airfoils per row is made possible by the use of a flexible database system. Results in the form of instantaneous pressure and entropy contours and time-averaged pressures are presented for the 2 1/2-stage compressor. Time-averaged pressures and pressure amplitudes for a single-stage turbine configuration are also presented. The numerical results compare well with experimental data.

Applications of a conservative zonal scheme to transient and geometrically complex problems

A conservative zoning technique, wherein the flow field for a finite-difference calculation is divided into several regions to simplify grid generation, is discussed and is applied in the solution of a two-dimensional problem of complex topology. Calculations are performed on two zonal, or patched, grid systems for the supersonic flow over a double-airfoil configuration. The solution is smooth and continuous across the zonal interfaces, and shock waves pass through the boundaries without distortion. In addition, the time accuracy of the zonal boundary method is verified by a two-zone cylinder calculation with a stationary inner and a rotating outer mesh. The feasibility of the zonal approach for use in the solution of geometrically complex and unsteady problems is thus demonstrated.

Direct Numerical Simulations of Transitional and Turbulent Flow on a Turbine Airfoil

A direct numerical simulation of transitional/turbulent flow on a low-speed turbine airfoil is presented here. The nonconservative form of the Navier–Stokes equations for compressible flows is used for this simulation. The numerical method used to solve these equations is a high-order-accurate upwind-biased iterative-implicit finitedifference scheme and is also presented here. The algorithm and the simulation are extensions of earlier efforts in direct simulations of transition and turbulence on flat plates. The present investigation has additional features, such as surface curvature, an adverse pressure gradient region on the airfoil, and trailing-edge vortex shedding. The results provided in the paper include the time-averaged pressure and the Stanton number distribution on the airfoil surface and turbulence statistics and flow visualization in the transitional/turbulent regions. Comparisons with experimental and computational data obtained for flows over flat plates and in channels are provided. The results indicate that the essential features of transition and turbulence have been captured.

Metric-discontinuous zonal grid calculations using the Osher scheme

Abstract Computations on zonal grids—in particular, grids with metric discontinuities resulting from the interspersion of highly clustered regions with coarse regions—are possible using a fully conservative form of the Osher upwind scheme. These zonal grids can result from an abrupt clustering of points near solution discontinuities or near other flow features that require improved resolution. The zonal approach is shown to capture shocks with almost “shock-fitting” quality but with minimal effort. Results for inviscid flow, including quasi-one-dimensional nozzle flow, supersonic flow over a cylinder, and blast-wave diffraction by a ramp, are presented. These calculations demonstrate the powerful capabilities of the Osher scheme used in conjunction with zonal grids in simulating flow fields with complex shock patterns.