Hybrid Reynolds-Averaged Navier-Stokes/Large Eddy Simulation Approach for Predicting Jet Noise

Abstract

Hybrid acoustic prediction methods have an important advantage over the current Reynolds-averaged NavierStokes based methods in that they only involve modeling of the relatively universal subscale motion and not the configuration-dependent larger-scale turbulence. Unfortunately, they are unable to account for the high-frequency sound generated by the turbulence in the initial mixing layers. This paper introduces an alternative approach that directly calculates the sound from a hybrid Reynolds-averaged Navier-Stokes/large eddy simulation flow model (which can resolve the steep gradients in the initial mixing layers near the nozzle lip) and adopts modeling techniques similar to those used in current Reynolds-averaged Navier-Stokes based noise prediction methods to determine the unknown sources in the equations for the remaining unresolved components of the sound field. The resulting prediction method would then be intermediate between the current noise prediction codes and previously proposed hybrid noise prediction methods. Nomenclature c = speed of sound D 0 = modified convective derivative D j = spatial derivative operator D Dt = convective derivative f = dummy variable g a = adjoint vector Green's function g = vector Green's function h = enthalpy h 0 = stagnation enthalpy K = component of linear Euler operator L = linear Euler operator p = pressurê p = component of pressure determined by base flow R = ideal gas constant R ij = residual stress tensor s = residual source vector T = absolute temperature, large time interval t = time u i = generalized residual velocity variable V = integration over all space v = velocity vector v i = component of v, i 1, 2, 3 x = position vector (usually associated with the observation point in the Green's function formulas) x i = Cartesian component of x, i 1, 2, 3 y = position vector of source y i = Cartesian component of y, i 1, 2, 3 jl = correlation of instantaneous propagator = specific heat ratio j = instantaneous propagator j = instantaneous propagator = five-dimensional Kronecker delta = Dirac delta function = separation vector in residual stress correlations ij = base flow stress tensor @ = generalized spatial derivative 00 = far-field pressure autocovariance = generalized residual pressure variable = densitỹ j = base flow stress tensor 0 j = generalized Reynolds stress tensor = source point time variablê = base flow quantity, filtered quantitỹ = base flow quantity, Favre-filtered quantity = time average hi = arbitrary spatial filter Superscripts 0 = residual quantity, dummy integration variable 00 = component of residual motion driven by residual stress, dummy integration variable, and fluctuating quantity

Sound propagation using an adjoint-based method

“…Adjoint Green's function for a source position at time , is defined by when instead of the physical problem defined in (2.1), an impulsive problem at position and time is considered: The general scalar reciprocity principle is recovered with help of Lagrange's identity (2.4) and the above-introduced scalar product : As the direct problem is causal, its adjoint is then necessary anti-causal, e.g. (Goldstein 2006, equation (A8)), This complies with the analysis of Karabasov et al (Semiletov & Karabasov 2013; Karabasov & Sandberg 2015) and Afsar et al (2017) who stressed that, the adjoint source acts as a sink and that the adjoint acoustic solution is inward-going, but goes beyond. The adjoint solution needs furthermore to go backward in time and is intrinsically anti-causal (Lamb Jr. 1995, §§ 4.5 and 4.6; Stone & Goldbart 2009, § 4.2).…”

The aeroacoustics of slowly diverging supersonic jets

“…But this may also be caused by the break down in the quasi-equilibrium assumption (on which the RANS modeling is based) in this region. These difficulties can probably be overcome by using a hybrid RANS/LES approach of the type described by Goldstein (2006), but that would greatly increase the expense of the computation.…”

“…. , 5 and can be inserted into Green's formula to obtain the following expression for the pressure-like variable p e (Goldstein 2006):…”

Recent Developments in the Application of the Generalized Acoustic Analogy to Jet Noise Prediction

“…Equations Goldstein (2006), with the subscript x, t being introduced to display its independent variables and u ν denotes the five dimensional solution vector…”