Turbulent thermal boundary layer on a permeable flat plate
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Turbulent Thermal Boundary Layer on a Permeable Flat Plate I. I. Vigdorovich Institute of Mechanics, Moscow State University, Moscow, 119192 Russia e-mail: [email protected] Received October 31, 2006
Abstract—Scaling laws are established for the profiles of temperature, turbulent heat flux, rms temperature fluctuation, and wall heat transfer in the turbulent boundary layer on a flat plate with transpiration. In the case of blowing, the temperature distribution represented in scaling variables outside the viscous sublayer has a universal form known from experimental data for flows over impermeable flat plates. In the case of suction, the temperature distribution is described by a one-parameter family of curves. A universal law of heat transfer having the form of a generalized Reynolds analogy provides a basis for representation of the heat flux distributions corresponding to different Reynolds numbers and transpiration velocities in terms of a function of one variable. The results are obtained without invoking any special closure hypotheses. PACS numbers: 44.20.+b, 47.27.nb, 47.27.te DOI: 10.1134/S1063776107060155
1. INTRODUCTION The dynamic problem for the turbulent boundary layer on a flat plate with transpiration was solved in [1−4]. The results were obtained under very general physical assumptions without invoking any special closure hypotheses, because a universal relation was found between turbulent shear stress and mean velocity gradient. The relation can be determined from the velocity profile in a reference flow (turbulent boundary layer on an impermeable flat plate). The resulting closed equations were solved by the method of matched asymptotic expansions in [5]. In this study, the approach developed in [2–4] is extended to the heat transfer problem. It is shown that its solution can also be determined from known profiles of velocity and temperature in the reference flow. Another extension is the use of a generalized boundary condition on the plate, where temperature follows an arbitrary law. In this case, the closure hypothesis relating the turbulent heat flux to the mean velocity and temperature gradients is a functional of the wall temperature distribution. Operator relations between dimensional physical quantities must have a special form determined by the invariance of the relations under the choice of units of measure. It is shown that the operator expression for a certain physical law can be represented as a functional of an invariant phase-space hypersurface, which reduces to a phase-plane curve when the law is expressed in terms of a function of one variable. In the problem under analysis, this approach can be used to determine the value of streamwise temperature
gradient along the plate that affects the solution. In particular, it is shown that the profiles of desired quantities corresponding to different boundary conditions (constant temperature or heat flux is specified on the wall) have similar representations in terms of scaling variables. 2. STA
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