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Week 5 - Turbulence modelling challenge

Turbulence is a phenomenon that is present in all flow seen around us and happens when the reynold's number of a flow exceeds the critical number. When turbulence occurs, eddies or vortices are formed in flow which drastically changes fluid properties in that area. Initially larger eddies are formed, which may break into…

Dushyanth Srinivasan
updated on 23 Jun 2022

Turbulence is a phenomenon that is present in all flow seen around us and happens when the reynold's number of a flow exceeds the critical number.

When turbulence occurs, eddies or vortices are formed in flow which drastically changes fluid properties in that area.

Initially larger eddies are formed, which may break into smaller eddies. Sometimes, smaller eddies can join together to join form larger eddies, this phenonmenon is called energy cascading.

Elementary properties of turbulence (Chapter 8) - Turbulence in Rotating, Stratified and Electrically Conducting Fluids

In a large eddy, the size is directly proportional to the ratio of length and freestream velocity. That is, $S i z e \propto \frac{L}{U_{\infty}}$ $Size prop L/U_(oo)$

In a small eddy, the size is directly proportional to the ratio of length and reynold's number. That is, $S i z e \propto \frac{L}{{(R e)}^{\frac{3}{4}}}$ $Size prop L/(Re)^(3/4)$

In order to quantify the scale of smaller eddies, Andrey Kolmogorov defined three relations quantifying the properties of small eddies, they are called Kolmogorov Microscales and are as follows:

Kolmogorov Length Scale

$η = {(\frac{ν^{3}}{ε})}^{1 / 4}$ $eta = (nu^3/epsilon)^(1//4)$

Kolmogorov Time Scale

$τ_{η} = {(\frac{ν}{ε})}^{1 / 2}$ $tau_eta = (nu/epsilon)^(1//2)$

Kolmogorov Velocity Scale

$u_{η} = {(ν ε)}^{1 / 4}$ $u_eta = (nu epsilon)^(1//4)$

Where $η$ $eta$ is the average rate of dissipation of turbulence kinetic energy per unit mass and is given by $η = \frac{U^{3}}{L}$ $eta = U^3/L$ and $ν$ $nu$ is the Kinematic Viscosity.

The ratio of largest to smallest eddy size is calculated by:

$\frac{L}{η} = \frac{L}{{(\frac{ν^{3}}{ε})}^{1 / 4}}$ $L/eta = L /((nu^3/epsilon)^(1//4))$

Substituting $ε$ $epsilon$ and simplyfying we get:

$\frac{L}{η} = {(\frac{L U}{ν})}^{3 / 4}$ $L/eta = ((LU)/nu)^(3//4)$

We know $\frac{L U}{ν} = R e$ $(LU)/nu = Re$

$\frac{L}{η} = R e^{3 / 4}$ $L/eta = Re ^(3//4)$

Turbulence Modelling is all about finding ways to account for small eddy behavior, using various methods and Kolmogorov Microscales.

They are three approaches for modelling these eddies, they are:

1. Direct Numerical Simulations (DNS)

2. Large Eddy Simulations (LES)

3. Reynold's Averaged Navier Stokes (RANS)

1. Direct Numerical Simulations (DNS)

In DNS, all eddies present in the flow are solved according to the Navier-Stokes equation. This method is extremely accurate but requires a lot of resources, since even the smallest eddies need to be divided amongst multiple elements.

Recalling the Kolmogorov Microscales,

$η = \frac{1}{{(R e)}^{3 / 4}}$ $eta = 1/(Re)^(3//4)$

The smallest volume for each element is given by: (in 3D)

$Δ V = η^{3} = \frac{1}{{(R e)}^{9 / 4}}$ $DeltaV = eta^3 = 1/(Re)^(9//4)$

The total number of elements is $\frac{V}{Δ} V$ $V/DeltaV$

Therefore, number of elements is $\propto R e^{9 / 4}$ $prop Re ^(9//4)$

In turbulent flow, $R e$ $Re$ is already high. This would bring the amount of elements required to astronomically high levels.

Hence, this method isn't feasible in commercial applications and is only used for research purposes.

2. Large Eddy Simulations (LES)

In LES, the smaller eddies are actively ignored to reduce simulation cost/resources, this leads to, as expected less accurate results. This model may be used for a baseline simulation, before proceeding with a better model.

Smaller eddies can be caluclated to a greater extent by using the subgrid scale model, where elements are divided in regions where accurate data is required.

3. Reynold's Averaged Navier Stokes (RANS)

Since both LES and DNS are impractical for commericial applications, RANS is a common tool used in consumer grade and enterprise grade CFD solvers such as FLUENT, Star-CCM+ and CONVERGE.

This method used Reynold's Decomposition which refers to separation of the flow variable (like velocity $u$ ) into the mean (time-averaged) component ( $\bar{u}$ $bar u$ $\bar{u}$ ) and the fluctuating component ( $u'$ $u'$ $u ′ {displaystyle u^{prime }} `U$ ).

Turbulent time scale of the flucuating component is way smaller than the time averaged time scale.

$u (x, y, z, t) = \bar{u} (x, y, z) + u' (x, y, z, t)$ $u (x,y,z,t) = bar u (x,y,z) + u'(x,y,z,t)$

Where $\bar{u} (x, y, z) = \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} u (x, y, z) d t$ $bar u (x,y,z) = 1/underset(Int)(t) int_0^(underset(Int)(t)) u(x,y,z) dt$ .

$\frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} u' (x, y, z, t) d t = 0$ $1/underset(Int)(t) int_0^(underset(Int)(t)) u'(x,y,z,t)dt = 0$

The integration of any flucuating variable over time is always going to be zero. In other words, the value of time averaged flucuating variables is zero.

When RANS is applied to the continuity equation,

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$ $(del u)/(del x) + (del v)/(del y) = 0$

Decomposing the variables to time averaged and fluctuating values,

$\frac{\partial (\bar{u} + u')}{\partial x} + \frac{\partial (\bar{v} + v')}{\partial y} = 0$ $(del (baru+u'))/(del x) + (del (barv+v'))/(del y) = 0$

Applying time integration/time averaging,

$\frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial (\bar{u} + u')}{\partial x} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial (\bar{v} + v')}{\partial y} d t = 0$ $1/underset(Int)(t) int_0^(underset(Int)(t))(del (baru+u'))/(del x)dt + 1/underset(Int)(t) int_0^(underset(Int)(t))(del (barv+v'))/(del y) dt= 0$

Applying the integral to every term,

$\frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial \bar{u}}{\partial x} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial u'}{\partial x} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial \bar{v}}{\partial y} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial v'}{\partial y} d t = 0$ $1/underset(Int)(t) int_0^(underset(Int)(t))(del baru)/(del x)dt + 1/underset(Int)(t) int_0^(underset(Int)(t))(del u')/(del x)dt +1/underset(Int)(t) int_0^(underset(Int)(t))(del barv)/(del y) dt + 1/underset(Int)(t) int_0^(underset(Int)(t))(del v')/(del y) dt= 0$

We know time integral/time averaging over a flucuating variable is zero,

$\frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial \bar{u}}{\partial x} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial u'}{\partial x} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial \bar{v}}{\partial y} d t + \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} \frac{\partial v'}{\partial y} d t = 0$ $1/underset(Int)(t) int_0^(underset(Int)(t))(del baru)/(del x)dt + cancel(1/underset(Int)(t) int_0^(underset(Int)(t))(del u')/(del x)dt )+1/underset(Int)(t) int_0^(underset(Int)(t))(del barv)/(del y) dt + cancel(1/underset(Int)(t) int_0^(underset(Int)(t))(del v')/(del y) dt)= 0$

Resulting equation,

$\frac{\partial \bar{u}}{\partial x} \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} d t + \frac{\partial \bar{v}}{\partial y} \frac{1}{\underset{I n t}{t}} \int_{0}^{\underset{I n t}{t}} d t = 0$ $(del baru)/(del x)1/underset(Int)(t) int_0^(underset(Int)(t))dt + (del barv)/(del y)1/underset(Int)(t) int_0^(underset(Int)(t)) dt= 0$

Evaluvating the integral,

$\frac{\partial \bar{u}}{\partial x} + \frac{\partial \bar{v}}{\partial y} = 0$ $(del baru)/(del x)+ (del barv)/(del y)= 0$

This is the result of applying Reynold's-Averaged Navier-Stokes (RANS) Equation on the 2D Navier-Stokies continuity equation.

The continuity equation is time-averaged and this method can be used to efficiently model turbulent eddies in a simulation, with additional models. These additional models are:

1. First Order Model

This model uses 1st order partial differential equations to solve for reynold's stresses. This model is further divided into three models:

Zero Equation Model: does not account for transient nature of turbulence and hence is inaccurate.
One Equation Model: assumes turbulent flow and laminar flow are analogous.
Two Equation Model: turbulent KE and turbulent dissipation rate are calculated, mostly commonly used in industries.

2. Second Order Model

This model uses 2nd order partial differential equations to solve for reynold's stresses. This model is further divided into two models:

Algebaric Stress Model: Most accurate but computationally extensive
Reynold's Stress Model

A simulation on a golf ball will be performed in ANSYS Fluent to study the behavior and effects of wake regions.

Geometry

The geometry was imported into DesignModeller, it consited of an enclosure with a half dimpled golf ball, that is a ball with a smoother lower hemisphere, and an upper hemisphere with dimples on the surface.

This is the geometry seen in DesignModeller:

Mesh

The default mesh was used, with the default sizing.

This is the final mesh:

Through a section plane in the middle,

The mesh has adaptive sizing, that is more elements are generated near the dimples of the half golf ball to capture the curvature more effectively.

Mesh Metrics

Since most elements have a quality greater than 0.7, it can be assured that the mesh quality is satisfactory for this simulation.

The mesh contains 85517 nodes and 453826 elements.

Simulation Setup

General

The simulation used a pressure-based, steady state solver.

Turbulence Model

The turbulence model used was k-omega SST as it is the preffered model for external flow simulation.

Boundaries

inlet: velocity-inlet with a magnitube of 60m/s, normal the boundary.

outlet: pressure-outlet with a gauge pressure of 0Pa.

top, bottom, front and back: symmetry boundary condition.

wall: wall boundary condition.

Intialisation and Calculation

This setup was hybrid initialised and calculated for 200 iterations, or until residuals dropped below 1e-3.

Results

Residuals

This was taken in Fluent.

The simulation took 102 iterations, and since residuals have dropped below 1e-3 and will continue to drop the solution can be called converged since no further significant change will take place.

Velocity Contour

This was taken in CFD-Post.

A low velocity region is formed behind the dimples and not the smooth surface, this is due to formation of turbulence near the dimples, due to their disruptive geometry while the lower smoother surface does not offer such disruption.

Pressure Contour

This was taken in CFD-Post.

A high pressure region is formed on the front of the ball due to accumulated air. Two low pressure regions are formed on the top and bottom of the ball, but the pressure in the bottom is significantly lower than the pressure in the top. This is due to dimples offering resistance to flow, which relatively increases air pressure in that region.

Velocity Vectors

This was taken in CFD-Post.

Note: Background contour is a velocity contour

From this contour and vector plot, it can be observed that air flows past the smooth surface more easily than the dimpled surface. The dimpled surfaces are very good at creating turbulence near the surface which impedes the smooth flow of air.

Due to differential in pressure drag experienced by the ball, the ball experiences more downward force than a standard fully dimpled golf ball.

Since air wraps around the smoother surface much further than the dimpled surface, the resulting moment will cause the ball to rotate in the counter clockwise direction.

Combining these effects, the ball will curve in a downward direction mid flight.

References

1. https://www.cambridge.org/core/books/abs/turbulence-in-rotating-stratified-and-electrically-conducting-fluids/elementary-properties-of-turbulence/4A2B8FEAC69150D082EDA4EA83626972

2. https://skill-lync.com/student-projects/week-8-literature-review-rans-derivation-and-analysis-100 (my previous literature review on RANS)