Predicting shock dynamics in the presence of uncertainties

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Abstract

We revisit the classical aerodynamics problem of supersonic flow past a wedge but subject to random inflow fluctuations or random wedge oscillations around its apex. We first obtain analytical solutions for the inviscid flow, and subsequently we perform stochastic simulations treating randomness both as a steady as well as a time-dependent process. We use a multi-element generalized polynomial chaos (ME-gPC) method to solve the two-dimensional stochastic Euler equations. A Galerkin projection is employed in the random space while WENO discretization is used in physical space. A key issue is the characteristic flux decomposition in the stochastic framework for which we propose different approaches. The results we present show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness. However, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances. The multi-element version of polynomial chaos seems to be more effective and more efficient in stochastic simulations of supersonic flows compared to the global polynomial chaos method.

Introduction

Studies of noisy supersonic flow past aerodynamic objects are very limited. On the experimental side, the presence of inherent noise in high-speed wind tunnels make them unreliable predictors of fight performance, except in special cases [1]. In flight conditions, the sources of such “noise” may be different; for example, random disturbances may arise due to turbulence or due to other local small-scale events such as microbursts. However, even at subsonic conditions in unsteady flows past bluff objects, small amounts of noise may have a potentially dramatic effect on the structure of the mean flow. This was demonstrated in [2] for a uniform but noisy flow past an oscillating circular cylinder. In the absence of any disturbances a vortex street is formed in the wake characterized by the shedding of three vortices per shedding cycle, the so-called (P + S) pattern [3]. However, when noise is introduced at the inflow the vortex street switches to another state above a certain noise threshold, characterized by only two vortices per shedding cycle, as in the standard von Karman street for stationary cylinders (2S pattern). This phenomenon is discussed in detail in [2] and independent experimental validation can be found in [4].

For supersonic flows, random inflow disturbances may also modify substantially the flow field rendering traditional predictive CFD tools invalid. We demonstrate here such global flow structure change with a deterministic example. We consider supersonic flow (Mach number 2) past a wedge of angle θ = 14.7° subject to a small time-dependent inflow variation superimposed on a uniform inflow. Specifically, the two-dimensional velocity field is given byu=u1cosθ(1+ϵsin(ωπt)),v=-u1sinθ(1+ϵsin(ωπt)),where the notation, solution algorithms and a sketch of the set-up are explained in detail in the next two sections. Here, we plot the streamlines of the flow and the shock in Fig. 1 for ϵ = 0.06 and ω = 1. We see that both the streamlines and the shock are visibly perturbed with respect to the familiar classical “clean inflow” picture [5].

In this paper, we study the shock dynamics in two-dimensional supersonic flows past a wedge assuming two different random disturbances. These correspond to random inflow velocity or random oscillations of the wedge around its apex and they can be steady in-time or time-dependent. The latter could be a model for aeroelastic motions, e.g. for the flutter oscillations. Our goal is to develop robust predictive CFD tools to simulate these cases and evaluate these new tools. To this end, we will also develop analytical stochastic solutions for the wedge problem in the spirit of our previous work on the stochastic piston problem [6].

Specifically, we are interested in applying polynomial chaos to compressible flow problems. This method, first introduced by Ghanem and co-workers for various problems in mechanics [7], [8], models uncertainty by a spectral expansion based on Hermite orthogonal polynomials in terms of Gaussian random variables. A broader representation but within the same Galerkin framework as in [7], called “generalized polynomial chaos”, was introduced in [9], [10]. This version employs a broad family orthogonal polynomials from the Askey scheme as the expansion basis to represent non-Gaussian processes more efficiently; it includes the classical Hermite polynomial chaos as a subset. This version can be thought of as a counterpart of the global spectral method applied in simple-geometry deterministic PDEs. We also study in the current work a multi-element version of generalized polynomial chaos, similar to spectral/hp element method, but decomposing the multi-dimensional random space in a non-conforming fashion.

The paper is organized as follows: In Section 2, we derive simple analytical stochastic solutions, and in Section 3, we present the numerical methodology. In Section 4, we present numerical results from stochastic simulations based on the aforementioned two versions of polynomial chaos but also on Monte-Carlo simulations. We conclude in Section 5 with a brief discussion. In Appendix A we give details of the Galerkin projection and the derived modified Euler equations obtained using generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC).

Section snippets

Stochastic analytical solutions

We consider the perturbation of an oblique attached shock in supersonic flow past a wedge due to time-varying random inflow or random wedge motions. A schematic of this problem is shown in Fig. 2. We consider small perturbations and we also assume that the perturbation of the shock slope is small. The flow between the shock and the wedge is approximated as isentropic.

We denote the wedge angle by θ, the shock angle by χ and the incoming flow velocity W1 with its normal component u1 = W1 sin χ. The

Numerical methods

We solve the two-dimensional Euler equations for supersonic flow past a wedge for the two aforementioned cases: (1) random inflow, and (2) random wedge oscillations. In the latter case, we employ a transformation based on a boundary-fitted coordinate system approach so that we solve the Euler equations in a stationary domain. In order to compare differences with the analytical solutions we will perform two types of stochastic simulations following a Monte-Carlo approach and a polynomial chaos

Stochastic simulations

We now present representative results for the following conditions with respect to the boundary-fitted coordinate, as shown in Fig. 2. The length of the wedge is 5 while the angle of the shock is χ = 45°, the angle of the wedge is θ = 14.7436°, and the angle between the shock and the wedge is α = 30.2564°. The inflow Mach number M1 = 2, the inflow x velocity component is u1 = 1.9342 and the y velocity component is v1 = −0.509. Also, the sound speed is C1 = 1, the pressure p1 = 1, and the density ρ1 = 1.4. On the

Summary and discussion

True prediction of the shock dynamics in supersonic flows implies that uncertainties of many types, e.g. due to boundary conditions, geometric regularity, transport coefficients, etc., should be modeled properly and their effect be propagated accurately through the nonlinear flow equations and not simply as an afterthought. This, in turn, implies that we have to reformulate the Euler equations within the stochastic framework thereby expanding the dimensionality of the problem. In addition, we

Acknowledgments

This work was supported by the Computational Mathematics Program of AFOSR and by the Division of Mathematical Sciences of NSF. Computations were performed at NCSA, NPACI and PSC.

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