Week 8: Literature review RANS derivation and analysis
Week 8: Literature review RANS derivation and analysis
Aim:
- To apply Reynolds decomposition to the Navier-Stokes equations and come up with the expression for Reynolds stress.
- To explain the understanding of the terms Reynolds stress.
- To explain about turbulent viscosity and it's different from molecular viscosity.
Theory:
Turbulence:
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids.
The turbulence length scale, is a physical quantity describing the size of the large energy-containing eddies in a turbulent flow. The turbulent length scale is often used to estimate the turbulent properties on the inlets of a CFD simulation.
Reynolds Number:
The Reynolds number is the ratio of inertial forces to viscous forces. The Reynolds number is a dimensionless number used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid. Mathematically, the Reynolds number is given as,
`Re = (rhovd)/mu`
Where `rho` is density, v is velocity, d is diameter and `mu` is viscosity.
The Reynolds number is used to determine whether a fluid is in laminar or turbulent flow. For example, for the flow in a round pipe, the characterisation is given by
- Reynolds number < 2300 is Laminar flow
- Reynolds number between 2300 and 4000 is Transient flow
- Reynolds number > 4000 is Turbulent flow
Turbulence modelling:
Turbulence is a type of fluid flow that is unsteady, enormously irregular in space and time, three-dimensional, rotational and dissipative in terms of energy and diffusive transport phenomenon at high Reynolds numbers. Considering the properties of turbulent flow, an extremely small-scale fluctuations emerge in velocity, pressure, and temperature. Despite the fact that direct implementation of fluctuated values into the Navier-Stokes equation is possible but it requires an extreme amount of resources in terms of hardware, software, and human effort. Therefore, an appropriate numerical model should be implemented when modelling turbulent flow.
The main purpose of turbulence modelling is to prompt equations to anticipate the time-averaged velocity, pressure, and temperature fields, without calculating the complete turbulent flow pattern as a function of time as in RANS and LES.
Turbulence Modelling Approaches:
There are two approaches the CONVERGE provides. They are,
1. Reynolds Averaged Navier-Stokes (RANS) - The main idea with RANS is we solve equations for an ensemble averaged velocity field and the magnitude of the turbulent fluctuations.
2. Large Eddy Simulation (LES) - The LES solves the equation for a filtered time-dependent velocity field that represents large scale turbulence motion.
Reynolds Decomposition:
The Reynolds averaged Navier-stokes equations (RANS equation) are time-averaged equations of motion for fluid flow. The idea behind the equation is Reynolds decomposition whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities. The objective is to apply the decomposition process to Navier Stokes equations and derive an expression for Reynolds's stress. The RANS equation is primarily used to approximate the turbulent flow. To apply the Reynolds decomposition to the Navier-stokes equation, we need to substitute the Reynolds decomposition method by introducing time-averaged values and fluctuating values into the basic Navier-stokes equation.
`u_i = bar(u_i) + u_i ^'`
Inserting the above equation into the Navier-Stokes equations and ensemble averaging the result yields the RANS equation. The RANS equations contain an additional term called the Reynolds Stress Tensor which represents the effects of modelled turbulence.
Navier-Stokes equations:
The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum equation. The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made further approximations and simplifications to the equation set until they had a group of equations that they could solve.
Continuity Equation:
`(delu)/dx+(delv)/(dely)=0`
Momentum Equation:
`rho[(delu)/(delt)+u*(delu)/(delx)+v*(delu)/(dely)]=-(delrho)/(delx)+mu*(del^2u)/(dely^2)`
Energy Equation:
`u(delT)/(delx)+v(delT)/(dely) = alpha[(del^2T)/(delx^2)+(del^2T)/(dely^2)]`
Reynolds Decomposition on Navier-Stokes Equation:
Derivation on Continuity Equation:
`(delu)/dx+(delv)/(dely)=0`
By applying the Reynolds decomposition,
`(del(baru+u^'))/(delx)+(del(barv+v^'))/(delv)`
Time averaging the above equation we get,
`1/t_iint_0^(t_i)[(del(baru+u^'))/(delx)*dt+(del(barv+v^'))/(dely)*dt]=0`
`1/t_i[int_0^(t_i) (delbaru)/(delx)*dt+int_0^(t_i)(delu^')/(delx)*dt+int_0^(t_i)(delbarv)/(dely)*dt+int_0^(t_i)(delv^')/(dely)*dt]=0`
The time integration of fluctuating terms will be zero. Then the final equation is
`1/t_i[int_0^(t_i) (delbaru)/(delx)*dt+int_0^(t_i)(delbarv)/(dely)*dt]=0`
`1/t_i[(delbaru)/(delx)*t_i+(delbarv)/(dely)*t_i]=0`
`(delbaru)/(delx)+(delbarv)/(dely)=0`
From the above equation it is clear that Reynolds decomposition satisfies the Continuity equation.
Derivation on Momentum Equation:
`rho[(delu)/(delt)+u*(delu)/(delx)+v*(delu)/(dely)]=-(delrho)/(delx)+mu*(del^2u)/(dely^2)`
On multiplying `1/rho`on both sides
`[(delu)/(delt)+u*(delu)/(delx)+v*(delu)/(dely)]=-(1)/(rho)(delrho)/(delx)+(mu)/(rho)*(del^2u)/(dely^2)`
Now we are adding the x-component of velocity multiplied by continuity equation on left hand side for simplification purpose. In the above derivation continuity equation is satisfied which states that the results of these adding terms will be zero. Hence the balance of the momentum equation is not disturbed.
`(delu)/(delt)+u*(delu)/(delx)+v*(delu)/(dely)+u[(delu)/(delx)+(delv)/(dely)]=-(1)/(rho)(delrho)/(delx)+(mu)/(rho)*(del^2u)/(dely^2)`
After doing this simplification on application of product rule the equation changes to,
`(delu)/(delt)+2u*(delu)/(delx)+(del(u*v))/(dely)=-(1)/(rho)(delrho)/(delx)+(mu)/(rho)*(del^2u)/(dely^2)`
by applying `2u*(delu)/(delx)=(delu^2)/(delx)`
`(delu)/(delt)+*(delu^2)/(delx)+(del(u*v))/(dely)=-(1)/(rho)(delrho)/(delx)+(mu)/(rho)*(del^2u)/(dely^2)`
By applying Reynolds decomposition,
`1/t_iint_0^(t^1)[(del(baru+u^'))/(delt)*dt+(del(baru+u^')^2)/(delx)*dt+((del(baru+u^')*(barv+v^'))/(dely)*dt)]=1/t_iint_0^t-i[-1/rho((del(barp+p^'))/(delx)*dt+mu/rho*(del^2(baru+u^'))/(dely^2)*dt)]`
On simplification the above equation,
`(delu)/(delt)+(delu^2)/(delx)+(del(baru*barv))/(dely)=-1/rho(delbarrho)/(delx)+mu/rho*(del^2baru)/(dely^2)-1/t_i int_0^(t_i) (delu^2)/(delx)*dt-1/t_i int_0^(t_i) (delu^'v^')/(dely)*dt`
The fluctuating terms in the right hand side is not equal to zero.
`-1/t_i int_0^(t_i) (delu^2)/(delx)*dt ne0`
`1/t_i int_0^(t_i) (delu^'v^')/(dely)*dtne0`
In general the average of the fluctuating terms will be zero but when we consider the first terms the square of any value can't be negative so it cannot be equal to zero. In the second term the multiplication of both component of velocities can't be predictable so we can't neglect that term. But when we are dealing with the terms along the Boundary layer the values along y-direction is more important than x-axis.
So the first term deals with the square of the x-component velocity, so it can be negligible because it shows a very minimum effect on the equation.
Therefore the final equation will be,
`(delu)/(delt)+(delu^2)/(delx)+(del(baru*barv))/(dely)=-1/rho(delbarrho)/(delx)+mu/rho*(del^2baru)/(dely^2)-1/t_i int_0^(t_i) (delu^'v')/(dely)*dt`
which can be written as,
`(delu)/(delt)+(delu^2)/(delx)+(del(baru*barv))/(dely)=-1/rho(delbarrho)/(delx)+1/rho*(del)/(dely)[mu*(delbaru)/(dely)-rho/t_i int_0^(t_i)(u^'v^')dt]`
Inertial terms:
`(delu)/(delt)+(delu^2)/(delx)+(del(baru*barv))/(dely)`
Pressure Term:
`-1/rho(delbarrho)/(delx)`
Molecular Viscosity or Viscous stress:
`mu*(delbaru)/(dely)`
Reynolds stress:
`rho/t_i int_0^(t_i)(u^'v^')dt`
Turbulent Viscosity:
The turbulent viscosity is also called as eddy viscosity. In the study of turbulence in fluids, a common practical strategy is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an effective viscosity, called the "eddy viscosity, which characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation), in contrast to the viscosity of the fluid itself, which must be positive by the second law of thermodynamics, the eddy viscosity can be negative.
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 ow to eddy viscosity in a turbulent flow.
Difference between Turbulent and Molecular Viscosity:
The Turbulent viscosity is an imaginary concept but molecular viscosity is a dynamic viscosity that will be present in all real fluids.The molecular velocity depends only on the properties of the fluid but the turbulent viscosity depends on the properties of the fluid, flow, adjacent boundary either fluid or solid.The Turbulent viscosity will only present in turbulent flow regions whereas molecular viscosity is present in all types of flows.