IV.2 Turbulence

Turbulence is virtually impossible to avoid in fluid flow problems, and its modelling is one of the most significant challenges in CFD. To understand why, let us estimate the computational cost of simulating turbulence. A turbulent flow is comprised of eddies on scales from the size of the domain down to a size called the Kolmogorov scale which is related to the viscosity of the fluid and the rate of dissipation of the turbulent energy

( )1/4 n3- j = e

Of course, such a flow obeys the NSE at all scales, and so we could simulate the flow. We would need to construct a mesh fine enough to resolve all these length scales. For the flow around a car, say at Re = 500, 000 (equivalent to the car travelling at about 20 km/hr),

~ 1mm, and so we would need

( ) L 3 12 -j ~ 10

cells, where L is a characteristic dimension of the car. This is well beyond the capacity of any existing computer. In general, the cost of DNS varies with Reynolds number as Re^9/4, so the more turbulent the flow the more expensive the simulation. We need to find some way to reduce the complexity of this calculation.

The way to do this is to split the flow into two components : an average component, and fluctuations around that average. The average component will depend on the exact problem being solved (geometry, boundary conditions etc) and so will be explicitly simulated, whilst the fluctuating components will be labelled ‘turbulence’, and we will replace these components with a simple model based on our knowledge of the properties of turbulence in general. For instance, the velocity can be written as

-- ' u-= u-+ u-

This averaging could be a time average, for example :

integral --- -1 t+dt ux = dt uxdt t

for the u_x component of velocity. Note that

--- --- u'x = 0 but u'x2/= 0

Of course,

u² is the expression for kinetic energy, so

u_x'² is going to be related to the kinetic energy of the turbulent component of the flow.

If we apply this averaging to the momentum equation

@u-+ \~/ .(u-u) = - 1 \~/ p + n \~/ 2u- @t r

we end up with a very similar equation called the Reynolds equation :

-- @-u- ---- 1- -- 2-- @t + \~/ .(u-u) + \~/ .R-= - r \~/ p + n \~/ u

containing an additional term $\~/$ .R. R is called the Reynolds stress, and it represents the effect of the fluctuating components on the mean flow. We can write it out in full :

( ----- ----- ----) ---- u'xu'x u'xu'y- u'xu'z R-= u'u'= u'yu'x u'yu'y u'yu'z u'u'- u'u'- u'u'- z x z y z z

Other than this additional term, the Reynolds equation is the same as the the original momentum equation, and can be solved in exactly the same way. Because it is an equation for the mean velocity, ignoring turbulent effects, the complexity of the flow that it describes is much less, and so the cost of the solution is also much less.

However we have a problem. In the original case we had 4 equations for 4 unknowns p, . Now we have 4 equations for more than 4 unknowns p, , . This is termed the closure problem - we need to find an expression for R in terms of p, somehow.

Turbulence is often seen to have a diffusive effect. For instance, perfume in still air diffuses away from its source. In turbulent air, the same effect occurs, but the mixing is much faster. Thus we will model the turbulence here as a diffusive term

\~/ .R = n \~/ 2u- -- turb --

where

_turb is the turbulent viscosity of the flow. This is known as the eddy viscosity, or Boussinesq approximation. If 1 velocity scale v_t and 1 length scale l_t can be used to describe the effects of turbulence, then

n = Cv l on dimensional grounds t tt

The turbulent kinetic energy gives a velocity scale

1/2 vt ~ k

The turbulent energy will also be dissipated continuously via viscous processes at the smallest scales (around the Kolmogorov scale). We can call the rate at which this happens the turbulent dissipation rate

has dimensions

, and so units m²s^-3. From this we can construct a length scale

3/2 k--- lt = e

Using all of this we have an expression for the turbulent viscosity

k2- nt = Cm e

Thus, if we can evaluate k and

, we have a model for $\~/$ .R and we can solve for the flow.

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