Turbulence With Pressure

We investigate the exact results of the Navier-Stokes equations using the methods developed by Polyakov. It is shown that when the velocity field and the density are not independent, the Burgers equation is obtained leading to exact N-point generating functions of velocity field. Our results show that, has to be generelized the operator product expansion, proposed by Polyakov both in the absence and the presence of pressure. We find a method to determine the extra terms in the operator product expansion and derive the it coefficients and find the first correction to probablity disturbuation function. In the general case and for small pressure, we solve the problem perturbatively and find the probablity disturbuation function for the Navier-Stokes equation in mean field approximation. PACS number 47.27.AK 1 1Introduction A theoretical understanding of turbulence has eluded physicists for a long time. A statistical theory of turbulence has been put forward by Kolmogorov [1], and further developed by others [2-4]. This is to model turbulence using stochastic partial differential equations. In this direction, Polyakov [5] has recently offered a field theoretic method for deriving the probability distribution or density of states in (1+1)-dimensional in the problem of randomaly driven Burgers equation. The importance of the Burgers equation is that, it is the simplest equation that resembles the analytic structure of the N-S equation, at least formally, within the scope of applicability of the Kolmogorov‘s arguments. Polyakov formulates a new method for analyzing the inertial range correlation functions based on the two important ingredients in field theory and statistical physics namely the operator product expansion (OPE) and anomalies. He argues that in the limit of high Reynold’s number because of existence of singularities at coinciding point, dissipation remains finite and all sublleading terms give vanishing contributions in the inertial range. By using the OPE one finds the leading singularities and can show that this approach is self-consistent. Here we generlize Polyakov‘s approach to the Burgers turbulence [5] in the presence of the pressure (i.e. the Navier-Stokes equation). We consider two distinct situations, firstly when the velocity field and the density are not independent, and secondly when the gradient of the pressure is small. In the first case we show that by a change of variables one can transform this problem to the Burgers equation and fllowing [6] we find the exact N-point generating function of the velocities. For the second case we calculate the effect of pressure perturbatively and find the assymptotic behavior of probablity disturbuation function in the first approximation. 2 On the other hand, using the results of [6], we calculate the OPE coefficiente proposed by Polyakov, and show that we have to correct the OPE both in the presence and the absence of the pressure. We write the generlized OPE and find the first correction to the PDF which was initially found by Polyakov. The paper is organised as follows. In section two we show that when the velocity and density are not independent the governing equation reduces to the Burgers equation. In section three we find the N-point generating functions for the velocity field and show that we have to generlize the OPE proposed by Polyakov. We find the OPE coefficients and derive the first correction to probablity disturbuation function. In section four we consider the effect of pressur perturbatively and find the probablity disturbuation function in the mean field approximation. 2The Compressible fluid and the Burgers equation We begin our discussion with the system describing compressible flows in an ideal (polytropic) gas ignoring dissipative effect [7], viz. ρt + vρx + ρvx = 0 (1) ρ(vt + vvx) + px = 0 (2) p = kρ , s = const. (3) Here ρ, v and p are the density, velocity field and pressure, respectively. The s is entropy, assumed to be constant, γ = Cp Cv , the ratio of specific heats and k is a constant. Also in inertial range, we ignore the driving force, which pumps energy in the large scale of system. The system of eqs.(1-3) has essentialy two variables (ρ and v), it is nonlinear and difficult to handel in complete generality. Nevertheless there has been considerable analytic 3 interest in this system. In particular, some progress can be made by seeking simple wave solutions where one of the dependent variables is a functions of the others. Rewriting eq.(13) in terms of v and ρ only by introducing the square of the speed of sound, a = ( ∂ρ )s=s0 = kγρ, we have: ρt + vρx + ρvx = 0 (4) vt + vvx + a(ρ) ρ ρx = 0 (5) Now, we assume that v = v(ρ), so that eqs.(4) and (5) changes to the following equations: ρt + (v + ρv′)ρx = 0 (6) ρt + (v + a ρv′ )ρx = 0 (7) Here the prime denotes differentiation with respect to ρ. This system of linear algebraic equations in ρt and ρx has a non-trivial solution provided the determinant of the coefficient matrix vanishes so that: v′ = ± a ρ = ±a0( ρ ρ0 ) γ−1 2 1 ρ (8) The system (6-7), then reduces to one of the equations: ρt + (v ± a)ρx = 0 (9) where v(ρ) = ∫ ρ ρ0 a(ρ) ρ dρ = 2 γ − 1 {a(ρ)− a0} (10) We restrict our attention to waves moving to the right thus choosing the plus sign in eqs.(8) and (9). The corresponding equation for v follows easily from multiplication of eq.(6) by