Turbulent Flows

What is Fluid Turbulence?

- Turbulence is a three-dimensional, unsteady, irregular condition of fluid motion. The mass, momentum and all the species fluctuate in time and space.

- Due to the fluctuations, the mixing of quantities is significant, dP and the heat transfer effect more.

- Statistical averages (in time) of the flow field provide meaningful information on the fluid motion.

- The fluctuations are due to the swirling flow structures (Eddies) with a wide range of sizes.

 

Transition from Laminar to Turbulent

- For any laminar flow there is a value of Re at which the flow transitions to turbulent flow.

- The laminar-turbulent transition in the boundary layer on a solid body is affected by;

  Re, pressure difference, roughness, and the level of disturbance in the flow (turbulence intensity)

- Near the edge of the plate, the boundary layer is initially laminar but the laminar flow is replaced by the Tollmen-Schlichting instability waves.

- For a plate with a sharp leading edge, the transition takes place at a distance x from the leading edge given by:

$$ Re_{x,c} = (\frac{U_{\infty}x}{\nu})_{c} = 3.5\times 10^{5} to 10^{6} $$

 

Concept of Stability and Tranasition

What triggers this transition?

Can a specified physical state withstand small disturbances and return to its original unperturbed state?

If the disturbances die away in time, the basic flow is considered to be stable; if they grow, the basic flow is unstable and a transition may occur.

$k - \omega$ SST model

There are the following topics:

1. How is the $k-\omega$ SST model different from the $k-\epsilon$ and $k-\omega$ models?

2. What is the blending function $F_{1}$?

3. What is the difference between the BST and SST models?

4. What is the viscosity limiter?

 

Problems with the $k-\epsilon$ and $k-\omega$ models

1. The near wall damping functions ($f$) in the $k-\epsilon$ model are unreliable in a variety of flows

2. The $k-\omega$ model is sensitive to the freestream value of $\omega$ that is applied at the inlet

3. The wall shear stress is too high and the flow does not separate from smooth surfaces

>>> The $k-\omega$ SST model attempts to address these problems and give better separation prediction

 

The $k-\epsilon$ model

The standard $k-\epsilon$ model is :

$$ \frac{\partial{\rho k}}{\partial t} + \nabla \cdot (\rho U k) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla k) + P_{k} - \rho \epsilon $$

 

$$ \frac{\partial{\rho \epsilon}}{\partial t} + \nabla \cdot (\rho U \epsilon) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla \epsilon) + C_{1 \epsilon}P_{k} \frac{\epsilon}{k} - C_{2 \epsilon} \rho \frac{\epsilon ^{2}}{k}$$

but $ \epsilon = C_{\mu}k\omega$ - Substituting to $k-epsilon$ model gives:

 

$$ \frac{\partial{\rho \omega}}{\partial t} + \nabla \cdot (\rho U \omega) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla \omega) + \frac{\gamma}{\nu_{t}} P_{k} - \beta \rho \omega^{2} + 2 \frac{\rho \sigma_{ \omega 2}}{\omega} \nabla k: \nabla \omega$$

 

Compare this to the $k-\omega$ model:

$$ \frac{\partial{\rho \omega}}{\partial t} + \nabla \cdot (\rho U \omega) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla \omega) + \frac{\gamma}{\nu_{t}} P_{k} - \beta \rho \omega^{2} $$

 

There's this additional term in the $k-\epsilon$ model:

$$ 2 \frac{\rho \sigma_{ \omega 2}}{\omega} \nabla k: \nabla \omega$$

If we multiply this by $(1-F_{1})$ we can blend between the $k-\omega $ and $k-\epsilon$ models:

$$ 2 (1-F_{1})\frac{\rho \sigma_{ \omega 2}}{\omega} \nabla k: \nabla \omega$$

when $F_{1} =0$ the model is $k-\epsilon$

when $F_{1} =1$ the model is $k-\omega$

The $k-\omega$ BST equations

The transport equation for the $ k- \omega $ BST models are:

$$ \frac{\partial(\rho k)}{\partial t} + \nabla \cdot (\rho U k) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla k) + P_{k} - \rho \epsilon $$

 

$$ \frac{\partial(\rho \omega)}{\partial t} + \nabla \cdot (\rho U \omega) = \nabla \cdot ((\mu + \frac{\mu_{t}}{\sigma_{k}})\nabla \omega) + \frac{\gamma}{\nu_{t}}P_{k} - \beta \rho \omega ^{2} + 2(1-F_{1}) \frac{\rho \sigma_{\omega 2}}{\omega} \nabla k : \nabla \omega $$

where

$$ \nabla k: \nabla \omega = \frac{\partial k}{\partial x_{j}}\frac{\partial \omega}{\partial x_{j}} = \frac{\partial k}{\partial x}\frac{\partial \omega}{\partial x}+\frac{\partial k}{\partial y}\frac{\partial \omega}{\partial y}+\frac{\partial k}{\partial z}\frac{\partial \omega}{\partial z} $$

The Viscosity Limiter

The $k-\omega$ BST model still overpredicts the wall shear stress. The viscosity limiter is used to extend the BST model to the  SST model.

$$ \mu_{t} = \frac{\rho k}{\omega} ; Original$$

$$ \mu_{t} = \frac{a_{1}\rho k}{max(a_{1}\omega, SF_{2}} ; SST model$$

This limiter results in better agreement with experimental measurements of separated flow.

$F_{2}$ is another blending function that is similar to $F_{1}$

If $F_{2}$ or $S$ is large, then the viscosity is limited.

 

The $k-\omega$ model gives better agreement with experiments of mildly separated flows due to the viscosity limiter.

It is best for external aerodynamics or simulations where separation is important.