Session D27: Transonic and Turbomachinery CFD

Transonic Flows of Bethe-Zel'dovich-Thompson Fluids

We examine steady transonic flows of Bethe-Zel'dovich-Thompson (BZT) fluids over thin turbine blades or airfoils. BZT fluids are ordinary fluids having a region of negative fundamental derivative over a finite range of pressures and temperatures in the single phase regime. We present the transonic small disturbance equation, shock jump conditions, and shock existence conditions capable of capturing the qualitative behavior of BZT fluids. The flux function is seen to be quartic in the pressure or density perturbation rather than the quadratic (convex) flux function of the perfect gas theory. We show how this nonconvex flux function can be used to predict and explain the complex flows possible. Numerical solutions using a successive line relaxation (SLR) scheme are presented. New results of interest include shock-splitting, collisions between expansion and compression shocks, two compressive bow shocks in supersonic flows, and the observation of as many as three normal stern shocks following an oblique trailing edge shock.   

Supersonic flows of a BZT fluid over thin airfoils

We solve a quartic Burgers equation to describe the steady, two-dimensional, inviscid supersonic flow field of a Bethe-Zel'dovich-Thompson (BZT) fluid generated by thin airfoils or turbine blades. A parabolic arc airfoil has been considered. A motivation for this problem is to illustrate the complex flow patterns possible for simple airfoil shapes. The freestream state will be chosen so that the fundamental derivative of gas dynamics is negative for part or even all of the flow. The Burgers equation is solved using the WENO technique. This is the second motivation for this work to demonstrate that the WENO technique is well-suited to the study of BZT fluids. Phenomena of interest include the partial and complete disintegration of compression shocks, the formation of expansion shocks, and the collision of expansion and compression shocks.   

Entropy Generation in Three-dimensional, Swirling Flows

Entropy generation in a transonic, highly loaded, axial turbine stage has been investigated via application of the entropy transport equation. The use of a RANS simulation required additional consideration of entropy transport associated with the time-averaging of products of fluctuating quantities. Analysis of the RANS entropy transport equation allowed the investigators to calculate the volumetric distribution of $\overline{D}\overline{s}/\overline{Dt}$; that is, the time rate of change of mean entropy of a material particle. The quantity $\overline{D}\overline{s}/\overline{Dt}$ revealed localized regions of entropy increases and decreases of material particles. It was discovered that regions of large +$\overline{D}\overline{s}/\overline{Dt}$ occur primarily on the blade suction surface. Additionally, it was found that the passage vortex core, typically associated with high entropy fluid, corresponded with regions of $\overline{D}\overline{s}/\overline{Dt}$ $\approx$ 0. This result suggests that entropy is generated on the blade suction surface and then collected and convected by the blade passage vortex.   

The Impact of Casing Geometry on a High Speed Compressor Rotor Passage Shock

The adiabatic efficiency of a high speed compressor is inextricably linked with the strength and structure of the rotor passage shock. It is well known that the geometry of the compressor casing can affect the topology of the rotor passage flow field as well as the strength and location of the rotor passage shock. However, partially due to the complexities of this unsteady, swirling, and compressible flow, the community lacks a physics-based understanding of this phenomenon. As a first step in gaining insight into this problem, single-passage RANS simulations of Notre Dame's Stage 04 rotor with four different casing geometries were conducted. The presentation will examine changes in the topology of the rotor passage flow occurring with changes in the casing boundary condition.   

Global stability analysis of a transonic flow over OAT15A airfoil

A transonic interaction between a shock wave and a turbulent boundary layer on a supercritical profile is numerically and theoretically investigated. If the angle of attack is small, RANS simulations converge towards a steady solution; beyond a critical value, the shock exhibits self-sustained oscillations, and the flow can be related to the so-called transonic buffet. Linear stability analysis indicates that for low angle of attack the flow is stable in a global framework. In this case, the noise amplifier behavior of the flow is investigated through a singular value decomposition of the global Resolvent, which highlights the frequency selection process typical of shock-wave/boundary-layer interactions. It will be shown that the shock behaves as a low-pass filter, and Kelvin-Helmholtz type instability are related to high-frequency unsteadiness. When increasing the angle of attack, an unstable eigenvalue appears and the unsteady behavior can be correctly represented by the unstable global mode, as shown by Crouch et al. JFM 2009. The mechanism that is responsible for buffet onset will be discussed, and comparisons between adjoint/direct global modes and optimal forcing/response will be performed.   

Mach number effects on compressible flow past a circular cylinder at high Reynolds number

Compressible flows past a circular cylinder are numerically investigated by using constrained large-eddy simulation (CLES) method at a Reynolds number of 10$^{6}$ and with Mach number varying from 0.5 to 0.95. Several mean and statistical quantities, including the drag coefficient, the separation angle, the skin friction coefficient, and the pressure fluctuations are calculated and analyzed. It is found that the separation location moves toward first and then away from the front stagnation point with the increasing Mach number. Further analysis reveals that the fluid flow in the boundary layer on the cylinder changes from turbulent to laminar when the Mach number exceeds a critical value of Mcr $\sim$ 0.65. In other words, laminar boundary-layer separation over the circular cylinder will be observed when the inlet Mach number lies in certain range under consideration. Such a phenomenon is believed to be due to the compressibility effects as Mach number increases.