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Chapter 24 Linear Stability Analysis of Compressible Channel Flow Over Porous Walls

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Chapter 24 Linear Stability Analysis of Compressible Channel Flow Over Porous Walls Chapter 24 Linear Stability Analysis of Compressible Channel Flow over Porous Walls Iman Rahbari and Carlo Scalo 24.1 Introduction In the present work we investigate the effects of porous walls modeled with a simple Darcy-like boundary condition derived as a particular case of linear acoustic impedance boundary condition (IBC) with zero reactance. The latter can be directly written in the time domain as 0 v0 p D R n (24.1) 0 v0 where R is the impedance resistance, and p and n are the fluctuating pressure and wall-normal component of the velocity at the boundary (positive if directed away from the fluid side), normalized with the base density and speed of sound (unitary dimensionless base impedance). The present investigation is limited to a local linear stability analysis (LSA) and is inspired by the hydro-acoustic instability observed by Scalo et al. [20], who performed numerical simulations of compressible channel flow over a complex three-parameter broadband wall-impedance [21]. In the latter, the effective wall permeability is a broadband function of frequency tuned to allow maximum transpiration only for the large-scale turbulent energy containing eddies. Despite the lack of the frequency-dependency in (24.1), the results shown in the present investigation provide a useful theoretical framework for the understanding of hydro-acoustic instabilities and a guide for future investigations in the manipulation of compressible boundary layer turbulence. The effects of porous walls on shear flows have been a topic of formidable research effort, especially in the low-Mach-number limit. Lekoudis [15] performed LSA of an incompressible boundary layer over two types of permeable boundaries: I. Rahbari () • C. Scalo School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2045, USA e-mail: [email protected] © Springer International Publishing Switzerland 2017 451 A. Pollard et al. (eds.), Whither Turbulence and Big Data in the 21st Century?, DOI 10.1007/978-3-319-41217-7_24 452 I. Rahbari and C. Scalo one modeling a perforated surface over a large chamber (with stabilizing effects); the second one, modeling pores over independent cavities (with negligible effects). Jiménez et al. [12] performed incompressible channel flow simulations over active and passive porous walls observing the creation of large spanwise-coherent Kelvin– Helmholtz in the outer layer responsible for frictional drag increase; surprisingly, the near-wall turbulence production cycle was not found to be significantly altered [11]. More recently, Tilton and Cortelezzi [22] investigated channel flow coated with finite-thickness homogeneous porous slabs. Two unstable modes were observed, one symmetric and one anti-symmetric, originating from the left branch of eigenvalue spectrum. Similar results have been found in the present investigation, which also supports the findings of Scalo et al. [20] who observed a complete reorganization of the near-wall turbulence, following the application of the tuned wall-impedance. One of the earliest works in LSA of compressible shear flow has been carried out by Malik [17], who investigated high-speed flat plate boundary layers up to freestream Mach numbers of 10. Duck et al. [10] performed a similar investigation in the case of viscous compressible Couette flow, revealing fundamental differences with respect to external flow cases. Hu and Zhong [8, 9] investigated supersonic viscous Couette flow at finite Reynolds numbers finding two unstable acoustic inviscid modes when a region of locally supersonic flow, relative to the phase speed of the instability wave, is present. Their structure was deemed consistent with two unstable modes shown by Mack [16], which are sustained by the acoustic interaction between the walls and the sonic line. Malik and co-workers [18]have also used modal and non-modal stability to study the effect of viscosity stratification on the stability of compressible Couette flow. Energy transfer from the mean flow to perturbations occurs at two locations: near the top wall and in the bulk of flow domain, associated with two different unstable modes. More recent efforts are focused on the transitional hypersonic boundary layer stability [3, 5, 23–25], where porous walls are used as means to delay transition via acoustic energy absorption (allowed by a frequency-selective permeability). The present work aims to inform future investigations of wall-bounded com- pressible boundary layers over porous walls. Comparison against previous LSA studies of channel and Couette flow is first discussed. Impermeable-wall laminar and turbulent channel flow data used as a base flow in the present study are then presented. Finally, the structure of the unstable modes triggered by the porous walls is discussed along with their predicted effect on the structure of near-wall turbulence. 24.2 Problem Formulation The fully nonlinear governing equations are first introduced, followed by their linearized counterpart, used for the present LSA. All quantities reported hereafter, including results from previous works are normalized with the channel’s half-width (or total height in the case of Couette flow), the bulk density (constant for channel flow), and the speed of sound, temperature, and dynamic viscosity at an isothermal wall. 24 Linear Stability Analysis of Compressible Channel Flow over Porous Walls 453 24.2.1 Governing Equations The dimensionless governing equations for conservation of mass, momentum, and total energy are, respectively, @ @ ./ C uj D 0 (24.2) @t @xj @ @ @ @ .ui/ C uiuj D p C ij C f1ı1i (24.3) @t @xj @xi @xi @ @ @ .E/ C uj .E C p/ D uiij C qj C f1u1 (24.4) @t @xj @xj where x1, x2, and x3 (alternatively x, y, and z) are, respectively, the streamwise, wall- normal, and spanwise coordinates, ui the velocity components in those directions. Thermodynamic pressure, density, and temperature are related by the equation of state p D 1T, where is the ratio of specific heats. E is the total energy per unit mass and Rea the Reynolds number based on the reference length, (either the channel’s half-width or wall-to-wall distance for Couette flow), and speed of sound at the isothermal wall temperature, which is related to the bulk Mach and Reynolds numbers via Reb D Mb Rea. The viscous stress tensor and conductive heat fluxes are Ä 2 1 @um 1 1 @T ij D Sij ıij ; qj D (24.5) Rea 3 @xm 1 Rea Pr @xj where Sij is the strain-rateÁ tensor, the dynamic viscosity, respectively, given by @ 1 uj @ui n Sij D C , D T where n D 0:75 is the viscosity exponent and Pr is the 2 @xi @xj Prandtl number. The complete set of governing equations (24.2)–(24.4) are solved for compress- ible turbulent channel flow with impermeable, isothermal walls in discretized form on a Cartesian domain with the structured code Hybrid, originally developed by Johan Larsson for numerical investigation of the fundamental canonical shock– turbulence interaction problem [13, 14]. The code has been scaled up to 1:97 million cores in recent remarkable computational efforts by Bermejo-Moreno et al. [2]. A solution-adaptive strategy is employed to blend high-order central polynomial and WENO schemes in presence of strong flow gradient [1, 19]. A fourth order discretization both in space and time has been adopted in the present work. Results from the high-fidelity simulations are used to generate the base flow (see Sect. 24.4.1) for the present LSA. 454 I. Rahbari and C. Scalo 24.2.2 Linearized Equations Decomposing a generic instantaneous quantity, a.x; y; z/ into a base state, A.y/, and a two-dimensional fluctuation, a0.x; y; t/, assuming ideal gas and retaining only the first order fluctuations yields the following equations for conservation of mass, streamwise and wall-normal momentum, and energy [17]:  à @p0 1 @T0 1 @u0 @p0 1 @T0 1 @v0 1 @T C CU C v0 D 0 (24.6) T @t T 2 @t T @x T @x T 2 @x T @y T 2 @y Â Ã Ä Â Ã @u0 @u0 @U 1 @p0 M @2u0 @2v0 @2u0 U v0 C C D C l2 2 C l1 C 2 @t @x @y T @x Rea @x @x@y @y  à  à 1 dM dT @u0 @v0 1 @M @2U @U @T0 C C C T0 C M dT dy @y @x M @T @y2 @y @y 1 @2M @T @U C T0 M @T 2 @y @y (24.7) Â Ã Ä Â Ã @v0 @v0 1 @ 0 M @2v0 @2 0 @2v0 U p u C D C 2 C l1 C l2 2 C @t @x T @y Rea @x @x@y @y  à  à (24.8) 1 @M @T0 @U 1 @M @T @u0 @v0 C C l0 C l2 M @T @x @y M @T @y @x @y Â Ã Ä @T0 @T0 @T 1 @p0 @p0 C U C v0 D . 1/ C U C @t @x @y T @t @x "  à ! # M @2 0 @2 0 2 @K @T @ 0 1 @K @2T 1 @2K @T 2 T T T 0 C 2 C 2 C C 2 C 2 T ReaPr @x @y K @T @y @y K @T @y K @T @y " #  à  Ã2 . 1/M @U @u0 @v0 1 @M @U CC C2 C C Rea @y @y @x M @T @y (24.9) n where lj D j 2=3, M D T , K D Cp M=Pr, where M, K, T , and U are the base dynamic viscosity, conductivity, temperature, and streamwise velocity. In the derivation of Eqs. (24.6)–(24.9) it was assumed that P D 1=. The base density is derived from the base pressure and temperature via the equation of state with gas constant equal to 1=, consistently with the normalization adopted.
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