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Week 8: Literature review RANS derivation and analysis
I. Aim: Literature review RANS derivation and analysis II. Introduction: TURBULENT FLOWS : Generally, a flow is differentiated between a laminar and a turbulent flow state. If the flow velocity is very small, the flow will be laminar, and if the flow velocity exceeds a certain boundary value, the flow becomes turbulent.…
Rajesh Simhadri
updated on 12 Mar 2022
I. Aim: Literature review RANS derivation and analysis
II. Introduction:
TURBULENT FLOWS :
Generally, a flow is differentiated between a laminar and a turbulent flow state. If the flow velocity is very small, the flow will be laminar, and if the flow velocity exceeds a certain boundary value, the flow becomes turbulent. The following shows this transition from a “well-ordered” laminar state to a seemingly stochastic and “chaotic” turbulent state for pipe flow. This experiment was first done by Reynolds around 1860. Reynolds had then shown that the transition from laminar to turbulent can be described by the dimension-less Reynolds-number (Re).
Re=V.Lν
with u being the “typical“ flow velocity, L being the characteristic length, and n being the kinematical viscosity. The characteristic length L is used for the description of fluid-transport processes (hydrodynamics, mass transport, heat transport, etc.) in so-called dimensionless index numbers, such as the Reynolds-number or Prandtl-number. It has the dimension of a length but describes the three-dimensional geometry of a reference system. In simplified terms, we apply for the characteristic length in the Reynolds number the diameter if there is a pipe flow or the water depth if we have an open channel flow.
Figure: Reynold’s experiment showing the transition from laminar to turbulent pipe flow, taken
from [Van Dyke, 1982].
A turbulent flow can also be characterized by the following properties, according to [Ferziger et al.,
1999] :
- The process of turbulence is highly unsteady so that e.g. the flow velocity at a given point is subject to great variance over time.
- Turbulence is a three-dimensional phenomenon.
- Vortices are an essential part of the flow, and the interaction of vortices and the so-called vortex stretching are basic mechanisms that increase and widen turbulence.
- The mixing processes due to common diffusion are often multiplied by turbulence.
- The contact between fluid balls with small and large motion moments also increases because of the strong mixing processes. The acting viscous forces then lead to a loss of kinetic energy, while the loss is translated into inner energy, i.e. heat energy is produced. This process is thus irreversible and dissipative.
- Newer examinations confirm the occurrence of so-called coherent structures. These are reproducible and deterministic processes, which have a big influence on the mixing processes.
There are two approaches to modelling turbulence, the deterministic and statistical approach:
Figure: Approaches of the turbulence description (according to Nikora, 2008).
Numerical Approach of calculating Turbulence in fluid flows:
Turbulent flows are characterized by four main features: diffusion, dissipation, three-dimensionality, and length scales. For the numerical calculation of turbulent flows, an averaging of the Navier-Stokes equations of motion is carried out. The averaging can be done with different dimensions.
• Averaging over time (assuming a statistically stationary flow) or averaging over a measurement (assuming constant boundary)
• Averaging over a space direction, in which the average flow does not vary.
The following three approaches are applied most frequently in the numerical calculation of turbulent flows:
1. Reynolds Averaged Navier-Stokes equations (RANS)
2. Large Eddy Simulation (LES)
3. Direct Numerical Simulation (DNS)
Reynolds Averaged Navier-Stokes equations (RANS)
The most frequent, but also most simple averaging is with respect to time. However, this approach is based on the assumption that the turbulent velocity fluctuations are distributed stochastically, meaning there is a constant mean flow. This averaging leads to the so-called Reynolds Averaged Navier-Stokes equations (RANS). Additional terms with new variables occur in these partial differential equations because of the averaging. Consequently, there are suddenly more variables than equations. In order to close the motion equation system, additional model equations or approximations have to be made, which express the variables as a function of the velocity field of the time-averaged flow. Setting up these model equations is a further research area with the name “turbulence modeling”. At the time averaging it can differ between eddy-viscosity models and Reynolds-stress models. In the former case, the turbulence is expressed by introducing an additional viscosity, the eddy viscosity. In Reynolds-stress models, the turbulence is considered through direct approaches to individual turbulent stress terms.
The time-averaging of the Navier Stokes (NS)equations and the continuity equation for incompressible fluids, the basic equations for the averaged turbulent flow will be derived in the following sections.
In order to be able to take a time average, the momentary value is decomposed into the parts mean value and fluctuating value. The following figure shows the time averaging of turbulent flow variable fluctuations.
Turbulent velocity fluctuation in pipe flow as a function of time [Ref: Fredsøe, 1990].
The momentary velocity component is u, the time-averaged value is named ˉuand the fluctuating velocity u′. With help of this definition, the decomposition for the flow variables can mathematically be written as
u=ˉu+u′,v=ˉv+v′,w=ˉw+w′,p=ˉp+p′ Eq.1
Also, for the density and the temperature:
ρ=ˉρ+ρ′,T=ˉT+T′, Eq.2
The chosen averaging method takes the mean values at a fixed place in space and is averaged over a time span that is large enough for the mean values to be independent of it.
u̅=1∆t∫t0+t1t0u.dt Eq.3
The time-averaged values of the fluctuating values are defined to be zero:
¯u′=0,¯v′=0,¯w′=0,¯p′=0 Eq.4
Firstly the continuity equation is averaged. If we substitute the expressions for the velocities from Eq.1:
∂ˉu∂x+∂u′∂x+∂ˉv∂y+∂v′∂y+∂ˉw∂z+∂w′∂z=0 Eq.5
The time-average of the above equation is written as:
¯∂ˉu∂x+∂u′∂x+∂ˉv∂y+∂v′∂y+∂ˉw∂z+∂w′∂z=0 Eq.6
The following mathematical rules are applied to simplify and reduction of the above equation Eq.6
The averaged derivatives of the fluctuations are also zero according to these rules so that the time-averaged
the continuity equation is:
∂ˉu∂x+∂ˉv∂y+∂ˉw∂z=0
Consider the advection term in momentum equation for X-direction as follows:
u∂u∂x+v∂u∂y+w∂u∂z=∂(u2)∂x+∂(uv)∂y+∂(uw)∂z-u(∂u∂x+∂v∂y+∂w∂z)
The above equation simplifies to:
u∂u∂x+v∂u∂y+w∂u∂z=∂(u2)∂x+∂(uv)∂y+∂(uw)∂z
The expressions for the decomposition of the velocities from Eq. 1 are now substituted into the
transformed Navier-Stokes equation and a time-average is done:
¯ρ{∂(ˉu+u′)∂t+∂(ˉu+u′)2∂x+∂(ˉu+u′)(ˉv+v′)∂y+∂(ˉu+u′)(ˉw+w′)∂z}=¯Fx-∂(ˉp+p′)∂x+μ(∂2(ˉu+u′)∂x2+∂2(ˉu+u′)∂y2+∂2(ˉu+u′)∂z2)
where Fx is not subject to turbulent fluctuation.
Application of the rules from the above described mathematical rules shows that among others the terms ∂(¯ˉuiuj′)∂xj,¯∂(ui′)∂t,¯∂(ui′)∂xj from the equation above can be reduced and the equation can be transformed to:
ρ(∂ˉu∂t+∂¯u⋅u∂x+∂¯u′⋅u′∂x+∂¯u⋅v∂y+∂¯u′⋅v′∂y+∂¯u⋅w∂z+∂¯u′⋅w′∂z)
=Fx-∂ˉp∂x+μ(∂2ˉu∂x2+∂2ˉu∂y2+∂2ˉu∂z2)
where the term ∂2ˉu∂x2+∂2ˉu∂y2+∂2ˉu∂z2 can be denoted by Δˉu
Further small transformations, for example, a repeated application of the product rule and the continuity equation to the advection term, lead to a form of the time-averaged NS equations for all three directions as:
ρ(∂ˉu∂t+ˉu∂ˉu∂x+ˉv∂ˉu∂y+ˉw∂ˉu∂z)=Fx-∂ˉp∂x+μΔˉu-ρ(∂¯u′u′∂x+∂¯u′v′∂y+∂¯u′w′∂z)
ρ(∂ˉv∂t+ˉu∂ˉv∂x+ˉv∂ˉv∂y+ˉw∂ˉv∂z)=Fx-∂ˉp∂y+μΔˉv-ρ(∂¯v′u′∂x+∂¯v′v′∂y+∂¯v′w′∂z)
ρ(∂ˉw∂t+ˉu∂ˉw∂x+ˉv∂ˉw∂y+ˉw∂ˉw∂z)=Fx-∂ˉp∂z+μΔˉw-ρ(∂¯u′w′∂x+∂¯v′w′∂y+∂¯w′w′∂z)
Or in tensor form:
ρ(DˉuiDt)=Fi-∂ˉp∂xi+μΔˉui-ρ(∂¯ui′uj′∂xj)
The term ∂¯ui′uj′∂xj represent the Reynold's stress due to time averaging of N-S equations.
To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the closure problem.
Turbulent Viscosity:
The turbulent transfer of momentum by eddies giving rise to internal fluid friction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place on a much larger scale.
Boussinesq proposed relating the turbulence stresses to the mean flow to close the system of equations. The Boussinesq hypothesis is applied to model the Reynolds stress term. A new proportionality constant , the turbulence eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models.
Refeences:
https://en.wikipedia.org/wiki/Turbulence_modeling
https://web1.eng.famu.fsu.edu/~dommelen/courses/flm/14/topics/turb/node2.html
http://home.iitk.ac.in/~gtm/turbulence/lecture26/26_3.htm
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