Week 8: Literature review - RANS derivation and analysis

                       RANS LITERATURE REVIEW: DERIVATION OF RANS EQUATONS FOR TURBULENT FLUID FLOWS

OBJECTIVE

To apply Reynolds Decomposition to NS Equations and obtain the expression for Reynold's Stress

INTRODUCTION

Turbulence or Turbulent flow is motion of fluid characterized by chaotic change in pressure and flow velocity, unlike laminar flow where fluid flows in parallel layers without disruption between them.It is caused by excessive kinetic energy in parts of fluid flow.

Characteristics of Turbulent Fluid Flow:

• Turbulent flow is chaotic.
• It is always 3D and cannot be 1D , 2D or steady as the instantaneous properties of flow changes at every location at every instant of time.
• Turbulent flow is highly diffusive as lots of eddies are formed in several sizes. The eddies act like molecules and give rise to turbulent transport properties. The scales of the eddies are very large compared to molecular scales.
• The additional diffusivity in turbulent flow is a function of Reynolds number unlike that in laminar flow.
• The friction factor ratios of turbulent to laminar flows may be 10 to 100. Thus ignoring the additional diffusivity can result in severe underdesign or overdesign based on the application.

RANS BACKGROUND

• The Navier Stokes equations cannot be used to characterize the turbulent flow. For this purpose, Reynolds' Averaged Navier Stokes (RANS) equations are derived which are obtained by averaging the Navier Stokes equations to account for the turbulence fluctuations in the fluid field.
• This is obtained by first splitting the velocity field into its corresponding mean and fluctuating components and then averaging the modified equations with time. For a laminar flow, the fluctuating components are equal to zero.

`u_{i}=\overline{u_{i}}+u_{i}^{\prime}`

where,`baru_i` denotes the mean part and `(u_i)prime` denotes the fluctuating part.

RANS EQUATION(S) DERIVATION

EQUATION OF CONTINUITY

`\frac{\partial u_{i}}{\partial x_{i}}=0 \rightarrow (1)`

MOMENTUM EQUATION

`\rho[\frac{\partial u_{i}}{\partial t}+u_{j} \frac{\partial u_{i}}{\partial x_{j}}]=-\frac{\partial p}{\partial x_{i}}+\mu[\frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}}] \rightarrow \(2)`

Applying the mean and fluctuating velocity decomposition to above equations:

From (1):

`\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{i}}=0 \rightarrow (3)`

From (2):

 `\rho[\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partialt}+(\overline{u_{j}}+u_{j}^{\prime}) \frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}}]=-\frac{\partial(\bar{p}+p^{\prime})}{\partial x_{i}}+\mu \frac{\partial^{2}(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j} \partial x_{j}}] \rightarrow (4)`

If we analyze only the second term on the LHS of Eqn. (4), we get

`(\overline{u_{j}}+u_{j}^{\prime}) \frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}}=\frac{\partial(\overline{u_{j}}+u_{j}^{\prime})(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}}-(\overline{u_{i}}+u_{i}^{\prime}) \frac{\partial(\overline{u_{j}}+u_{j}^{\prime})}{\partial x_{j}}`

From Eqn. (3), the second term on the RHS of the above equation vanishes and can be replaced in Eqn. (4), which gives us

`\rho[\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial t}+\frac{\partial(\overline{u_{j}}+u_{j}^{\prime})(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}}]=-\frac{\partial(\bar{p}+p^{\prime})}{\partial x_{i}}+\mu[\frac{\partial^{2}(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j} \partial x_{j}}] \rightarrow (5)`

Now, we will average Eqn. (3) and Eqn. (5) by following some basic rules of averaging as follows:

` $$\begin{array}{l} \overline{\bar{a}}=\bar{a} \\ \overline{a+b}=\bar{a}+\bar{b} \\ \overline{a \bar{b}}=\bar{a} \bar{b} \end{array}$$ `

Averaging Eqn. (3), we get

`\frac{\partial \overline{(\overline{u_{i}}+u_{i}^{\prime})}}{\partial x_{i}}=0 \Rightarrow \frac{\partial(\overline{u_{i}}+\overline{u_{i}^{\prime}})}{\partial x_{i}}=0`

Since the velocity component has been decomposed into its mean and fluctuating parts, the mean of the fluctuating parts can be taken as zero. Hence, we get

`\frac{\partial \overline{u_{i}}}{\partial x_{i}}=0 \rightarrow (6)`

Averaging Eqn. (5), we get

`\rho[\frac{\partial \overline{u_{i}}}{\partial t}+\frac{\partial \overline{(\overline{u_{i} u_{j}}+\overline{u_{i}} u_{j}^{\prime}+\overline{u_{j}} u_{i}^{\prime}+u_{i}^{\prime} u_{j}^{\prime})}}{\partial x_{j}}]=-\frac{\partial \bar{p}}{\partial x_{i}}+\mu \frac{\partial^{2} \overline{u_{i}}}{\partial x_{j} \partial x_{j}}`

`\Rightarrow \rho[\frac{\partial \overline{u_{i}}}{\partial t}+\frac{\partial \overline{u_{i} u_{j}}}{\partial x_{j}}+\frac{\partial \overline{u_{i}^{\prime} u_{j}^{\prime}}}{\partial x_{j}}]=-\frac{\partial \bar{p}}{\partial x_{i}}+\mu \frac{\partial}{\partial x_{j}}(\frac{\partial \overline{u_{i}}}{\partial x_{j}}) \rightarrow (7)`

Again, if we analyze only the second term in the LHS of the above equation, we get

`\frac{\partial \overline{u_{i} u_{j}}}{\partial x_{j}}=\overline{u_{i}} \frac{\partial \overline{u_{j}}}{\partial x_{j}}+\overline{u_{j}} \frac{\partial \overline{u_{i}}}{\partial x_{j}}`

From Eqn. (6), the first term in the RHS above vanishesand can be replaced in Eqn. (7), which gives us

`\rho[\frac{\partial \overline{u_{i}}}{\partial t}+\overline{u_{j}} \frac{\partial \overline{u_{i}}}{\partial x_{j}}]=-\frac{\partial \bar{p}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}[\mu \frac{\partial \overline{u_{i}}}{\partial x_{j}}-\rho \overline{u_{i}^{\prime} u_{j}^{\prime}}] \rightarrow (8)`

If we compare Eqn. (8) with Eqn. (2) for the instantaneous and the mean velocity field, we see that there is an additional term `\rho \overline{u_{i}^{\prime} u_{j}^{\prime}}` with the mean velocity field. These terms are part of the Reynolds' stress tensor defined as:

`\tau_{i j}=\rho \overline{u_{i}^{\prime} u_{j}^{\prime}}`

Hence, the solution of a turbulent flow field depends on calculating the Reynolds' stress tensor. For this purpose, different turbulence models are developed which can approximate the above tensor.

The stress tensor is hence divided into two parts:

Viscous stresses: `\mu \frac{\partial \overline{u_{i}}}{\partial x_{j}}`

Turbuleent stresses or Reynold's Stresses: `\tau_{i j}=\rho \overline{u_{i}^{\prime} u_{j}^{\prime}}`

REYNOLDS STRESSES

The component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equation to account for turbulent fluctuations in fluid momentum.

                                                                                                        OR

The net transfer of momentum across a surface in a turbulent fluid because of fluctuations in fluid velocity. Also known as eddy stress.

TURBULENT VISCOSITY

The coefficient that relates the average shear stress and rate of velocity gradient in the turbulent flow is called the turbulent viscosity. The viscosity in the turbulent flow is non-homogenous, and it varies in space.

The turbulent transfer of momentum by eddies giving rise to an internal fluid friction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place on a much larger scale. ... Eddy viscosity is a function of the flow, not of the fluid. It is greater for flows with more turbulence.

MOLECULAR VISCOSITY

Molecular viscosity is the same as viscosity. The Coefficient of Molecular Viscosity is the same value as dynamic viscosity. Molecular viscosity is the transport of mass motion momentum solely by the random motions of individual molecules not moving together in coherent groups. Molecular viscosity is analogous in laminar flow to eddy viscosity in turbulent flow.

DIFFERENCE BETWEEN TURBULENT AND MOLECULAR VISCOCITY

Eddie Viscosity is an imaginary concept. It does not exist as the molecular viscosity, which is a well defined transport coefficient as stated by the Kinetic Theory. The Eddie Viscosity hypothesis was posed for making the things simpler, in the sense that the turbulent Reynolds stresses (which are ugly and nonlinear in velocity perturbations) are simplified to be proportional to the gradients of the mean velocity, as happens in newtonian laminar flows with the viscous stresses. The coefficient of proportionality is termed the Eddie Viscosity, which far from being a constant or fluid property, is a magnitude dependant on the flow field and its solution. The Eddie viscosity hypothesis is inherently wrong, in that the Reynold stresses are in general not co-linear with the mean velocity gradients, as has being discovered by DNS solutions. However, the numerical methods stemming from this simplification (such as RANS methods) are low-time consuming and can be used, sometimes massively, by computational fluid dynamicists to obtain approximate solutions of turbulent flows.