Week 8: Literature review - RANS derivation and analysis
Aim:
Literature Review - RANS derivation and analysis
Objective:
- To apply Reynold's decomposition to the NS equations and come up with the expression for Reynold's stress.
- To explain the understanding of the terms Reynold's stress
- To explain what is turbulent viscosity? and How is it different from molecular viscosity?
Introduction:
The various forms of the Navier Stokes equation for a 2D incompressible fluid flow.
Continuity equation:
`(delu)/(delx) +(delv)/(dely) = 0`
Momentum equation:
`rho((delu)/(delt) +u(delu)/(delx) +v(delu)/(dely)) = -(delp)/(delx)+ mu((del^2u)/(dely^2) )`
Energy equation:
`rhoc[(delT)/(delt) + u(delT)/(delx) + v(delT)/(dely)] = k(del^2T)/(delx^2) + mu((delu)/(dely))^2`
These three equations can be used to solve both laminar & turbulent models.
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. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.
Turbulence Modelling:
Turbulence modelling is the construction and use of a mathematical model to predict the effects of turbulence. In computational fluid dynamics (CFD), even simple flows are difficult to compute. And, the computational complexity grows exponentially when accounting for turbulence: irregular fluid motions that span a wide range of scales in space and time. Selecting the right method for modelling turbulence physics is critical and confusing.
The Complexity of the Turbulence Model:
The complexity of different turbulence models may vary strongly depends on the details one wants to observe and investigate by carrying out such numerical simulations. Complexity is due to the nature of the Navier-Stokes equation (N-S equation). N-S equation is inherently nonlinear, time-dependent, three- dimensional PDE. Turbulence could be thought of as instability of laminar flow that occurs at high Reynolds numbers. Such instabilities origin form interactions between nonlinear inertial terms and viscous terms in the N-S equation. These interactions are rotational, fully time-dependent and fully three-dimensional. Rotational and three-dimensional interactions are mutually connected via vortex stretching. Vortex stretching is not possible in two-dimensional space. That is also why no satisfactory two-dimensional approximations for turbulent phenomena are available. Furthermore, turbulence is thought of as a random process in time. Therefore, no deterministic approach is possible. Certain properties could be learned about turbulence using statistical methods. These introduce certain correlation functions among flow variables. However, it is impossible to determine these correlations in advance. Another important feature of a turbulent flow is that vortex structures move along the flow. Their lifetime is usually very long. Hence certain turbulent quantities cannot be specified as local.
Classification of Turbulent Models:
Nowadays turbulent flows may be computed using several different approaches. Either by solving the Reynolds-averaged Navier-Stokes equations with suitable models for turbulent quantities or by computing them directly.
Direct Numerical Simulation:
- Calculates the exact turbulence solution, but is too computationally intensive to be practical.
- Numerically solves the full unsteady Navier-Stokes equations.
The DNS method is the most accurate method to model the turbulence flow but it requires turbulent time length scales, and hence the mesh is supposed to be so small that it is physically impossible to simulate the process using current numerical and computational capabilities. Even a simple cylinder flow simulation would require at least 8 million cells in this model.
Reynolds Averaged Navier Stokes Simulation:
- Good enough solution for many industrial flows.
- Time-averaged solution “smooths” turbulent eddies.
- Requires least computational power.
Large Eddy Simulation:
- Computes time-varying flow, but models’ sub-grid-scale motions.
The LES is a model that is in research and is also computationally expensive, but not as expensive as the DNS. They perform most of the flow energy calculations, hence responsible for high conservation of flow properties.
For solving most engineering problems, RANS models turn out to be the most practical ones. However, these methods do not reveal the flow of physics adequately and are not effective for complex flows. DNS and LES predict the flow physics better but are yet to be used regularly as engineering tools.
RANS:
The Reynolds Averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds. The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier-Stokes equations.
1. To apply Reynold's decomposition to the NS equations and come up with the expression for Reynold's stress.
Reynold's Decomposition:
To be able to take a time-average, the momentary value is decomposed into parts mean value and fluctuating value.
The momentary velocity component is u, the time-averaged value is named ū and the fluctuating velocity has notation u'. The decomposition can mathematically be written as:
`u = baru + u'`
`v = barv + v'`
`w = barw + w'`
`p = barp + p'`
The chosen averaging method takes the mean values at a fixed place in space and averaged over a time span that is large enough for the mean values to be independent of it.
`baru = 1/(DeltaT) int_(T_0)^(T_0 +T_1) udt`
The Integration of fluctuating velocity comes out to be zero,
`1/T int_0^T u'(x,y,z,t) = 0`
Reynold's Decomposition:
Split velocity components or any other flow variable into a mean component and a fluctuating component.
`u(x,y,z,t) = baru(x,y,z) + u'(x,y,z,t)`
`v(x,y,z,t) = barv(x,y,z) + v'(x,y,z,t)`
`w(x,y,z,t) = barw(x,y,z) + w'(x,y,z,t)`
Taking overbar as an averaging operator
`baru(x,y) = 1/T int_0^T u(x,y,t)dt`
`barv(x,y) = 1/T int_0^T v(x,y,t)dt`
Since `baru` is the averaging operator
`baru = 1/T int_0^T (baru +u')dt`
`baru = 1/T int_0^T barudt +1/T int_0^T u'dt`
So, `baru` is not a function of time
`baru = baru(1) + 1/T int_0^T u'dt`
So `1/T int_0^T u'dt = 0`
We also decompose the pressure
`p = barp +p'`
This decomposition is applied to Navier-Stokes equations to obtain Time-averaged form.
1. Continuity Equation:
`(delu)/(delx) +(delv)/(dely) = 0`
Applying decomposition
Substitute `u = baru +u'`
and `v = barv +v'`
`(del)/(delx)(baru+u') +(del)/(dely)(barv +v') = 0`
Taking time average of the whole equation
`1/T int_0^T(del)/(delx)(baru+u')dt +1/T int_0^T(del)/(dely)(barv +v')dt = 0`
`1/T int_0^T(del baru)/(delx)dt +1/T int_0^T(del u')/(delx)dt +1/T int_0^T(delbarv)/(dely)dt +1/T int_0^T(del v')/(dely)dt = 0`
Since, Time integral of fluctuating component is 0 & time-averaged value is still an average function of time, we get,
`(delbaru)/(delx)1/T int_0^Tdt +(delbarv)/(dely)1/T int_0^Tdt = 0`
By solving the integration
`(delbaru)/(delx)[(T-0)/T] +(delbarv)/(dely)[(T-0)/T] = 0`
`(delbaru)/(delx) +(delbarv)/(dely) = 0`
So, obtained continuity equation in terms of time-averaged quantities.
2. Momentum Equation:
`rho((delu)/(delt) +u(delu)/(delx) +v(delv)/(dely)) = -(delp)/(delx)+ mu(del^2u)/(dely^2)`
Dividing both side by `rho` ,we get
`(delu)/(delt) +u(delu)/(delx) +v(delv)/(dely) = -1/rho*(delp)/(delx)+ nu(del^2u)/(dely^2)`
Adding continuity equation inside this equation.
`u((delu)/(delx) +(delv)/(dely)) = 0`
`(delu)/(delt) +u(delu)/(delx) +v(delv)/(dely) +u((delu)/(delx) +(delv)/(dely)) = -1/rho*(delp)/(delx)+ nu(del^2u)/(dely^2)`
`(delu)/(delt) +2u(delu)/(delx) +v(delv)/(dely) +u(delv)/(dely) = -1/rho*(delp)/(delx)+ nu(del^2u)/(dely^2)`
Applying product rule
`(del(u*v))/(dely) =v(delu)/(dely) +u(delv)/(dely)`
Put it into the equation
`(delu)/(delt) +2u(delu)/(delx) +(del(u*v))/(dely) = -1/rho*(delp)/(delx)+ nu(del^2u)/(dely^2)`
`(delu)/(delt) +(del(u^2))/(delx) +(del(u*v))/(dely) = -1/rho*(delp)/(delx)+ (nu(del^2u))/(dely^2)`
Now applying Integration and Reynolds decomposition
Substitute `u = baru +u'`
and `v = barv +v'`
`1/T int_0^T(del)/(delt)(baru+u') dt +(del)/(delx)(baru+u')^2 dt +(del)/(dely)[(baru+u')*(barv+v')] dt = 1/T int_0^T-1/rho*(del(barp+p')) /(delx)+ (nu(del^2(baru+u')))/(dely^2)`
`1/Tint_0^T(delbaru)/(delt) dt + (delu')/(delt) dt + (delbaru^2)/(delx) dt + (delu'^2)/(delx) dt + (del(2*baru*u'))/(delx) dt + (del(baru*barv))/(dely) dt + (del(baru*v'))/(dely) dt + (del(u'*barv))/(dely) dt + (del(u'*v'))/(dely) dt = 1/T int_0^T -1/rho*(delbarp)/(delx) dt - 1/rho*(delp')/(delx) dt + (nu(del^2baru))/(dely^2) dt + (nu(del^2u'))/(dely^2) dt`
The fluctuating terms are equal to zero hence eliminating them.
`(delbaru)/(delt) + (delbaru^2)/(delx) +(del(baru*barv))/(dely) = -1/rho*(delbarp)/(delx) + (nu(del^2baru))/(dely^2) - 1/T int_0^T(delu'^2)/(delx) dt - 1/T int_0^T (del(u'*v'))/(dely) dt`
The above equation is similar to the original momentum equation except it is in time-averaged form and two additional terms on the RHS. The additional time integration terms in RHS are not equal to zero & can be proved, however, to simplify the model certain important assumptions are to be applied,
Assuming `- 1/T int_0^T(delu'^2)/(delx) dt = 0` considering velocities in x direction to be of negligible gradient, we get,
`(delbaru)/(delt) + (delbaru^2)/(delx) +(del(baru*barv))/(dely) = -1/rho*(delbarp)/(delx) + (nu(del^2baru))/(dely^2) - 1/T int_0^T (del(u'*v'))/(dely)`
On simplifying RANS equation
`(delbaru)/(delt) + (delbaru^2)/(delx) +(del(baru*barv))/(dely) = -1/rho*(delbarp)/(delx) + 1/rho(del)/(dely)[(mu(delbaru))/(dely) + rho/T int_0^T (u'*v') dt]`
In the above equation:
`(delbaru)/(delt) + (delbaru^2)/(delx) +(del(baru*barv))/(dely)` - Inertial Terms
`-1/rho*(delbarp)/(delx)` - Pressure Term
`(mu(delbaru))/(dely)` - Molecular Viscosity
`rho/T int_0^T (u'*v') dt` - Momentum Diffusivity of Turbulence
`(del)/(dely)[(mu(delbaru))/(dely)]` - Viscous Stress Term
`(del)/(dely)[rho/T int_0^T (u'*v') dt]` - Reynold's Stress
Reynold's stress:
The Reynold's stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.
The theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, Reynold's stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which may be thought of as an effective turbulent viscosity.
In fact, while much effort has been expended in developing good models for the Reynolds stress in a fluid, as a practical matter, when solving the fluid equations using computational fluid dynamics, often the simplest turbulence models prove the most effective. One class of models, closely related to the concept of turbulent viscosity, are the k-epsilon turbulence models, based upon coupled transport equations for the turbulent energy density `k` (similar to the turbulent pressure, i.e. the trace of Reynold's stress) and the turbulent dissipation rate `epsilon`.
Typically, the average is formally defined as an ensemble average as in statistical ensemble theory. However, as a practical matter, the average may also be thought of as a spatial average over some length scale, or a temporal average. Note that, while formally the connection between such averages is justified in equilibrium statistical mechanics by the ergodic theorem, the statistical mechanics of hydrodynamic turbulence is currently far from understood. In fact, the Reynolds stress at any given point in a turbulent fluid is somewhat subject to interpretation, depending upon how one defines the average.
What is turbulent viscosity? and How is it different from molecular viscosity?
Turbulent Viscosity:
The coefficient that relates the average shear stress and rate of velocity gradient in the turbulent flow is called turbulent viscosity. The turbulent transfer of momentum by eddies give rise to internal fluid friction on a large scale. Its also known as eddy viscosity. These are enormous in nature and to capture these to extreme details in near impossible, if done computationally it's very costly. Thus, we devise a way to capture the net effective mixing of eddies using viscosity models. Turbulent viscosity increases the shear rate and affects the velocity and heat transfer.
Molecular Viscosity:
It's a fluid property and is same as viscosity. Its coefficient is the same as dynamic viscosity. Molecular viscosity is the transport of mass motion momentum solely by random motions of individual molecules not moving together in coherent groups. Molecular viscosity is analogous in laminar flow to eddy viscosity in a turbulent flow. It can be construed like some sort of internal friction between two layers of fluid developed while flowing.
Difference between them:
1. The turbulent viscosity is an imaginary concept but molecular viscosity is dynamic that is present in all fluids.
2. Turbulent viscosity depends on fluid flow whereas molecular viscosity depends only on fluid properties
3. Turbulent viscosity is present only in turbulent flow regions but molecular viscosity is present in all types of flows.
4. Turbulent viscosity is non-linear & extremely hard to estimate whereas molecular viscosity is easy to measure.