Discuss the main differences between turbulence modeling based on RANS and LES, in terms of modelling aspects and also of computational effort.

EXERCISE 1

(2) Sketch the flow field around a Baseball, an airfoil of an airplane at cruising attitude and speed, a car and a ship. Provide an estimate for Reynolds number of all flow problems and indicate in your sketch, where the flow is turbulent

1.Baseball

Re<2300-flow is laminar

2300<Re<4000-Transition

Re>4000 -Turbulent flow

2. An airfoil

3. a ship

4. a car

(3) Explain what happens in the flow transition and which factors have on influence on it.

Flow patterns in a fluid depend on three factors: the characteristics (density viscosity compressibility) of the fluid, the speed of the flow and the shape of the solid surface.

Three characteristics of the fluid are of special importance: viscosing, density and compressibility. Viscosity is the amound of internal priction or resistance to flow. All gases are compressible, where as liquid are practically incompressible, that is they can not be scull(scale)ted into smaller volums.

Conservation – сохранение

Infinitesimal – бесконечно малый

-

– change of storage mass in the system

– dencity

– speed

– area

– an increment of time

;

vorticity – вихрение

rotational – переменный

Diffusion by vorticity bigger (much larger) molecular diffusion.

Flow turbulence is a change of flow from laminat to turbulence form.

(4)

Describe the most important properties of turbulent flows.

Irregularity. Turbulent flows are chaotic. That’s why, turbulence problems are treated statictically letter than deterministically.

Diffisivity. It is the characteristic, which provide enhanced mixing and increased rates of mass momentum and energy transports in a flow.

Rotationality. Turbulent flows have non-zero vorticity and are characterized by vortex stretching (a strong three dimensional vortex generation mechanism)

The stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the target flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid’s molecular viscosity into heat. This is why turbulence is always rotational and three-dimensional.

Dissipatior. To sustain turbulent slow a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress.

(5)

EXERCISE 2

A Dns is used to determine the decay of isotropic turbulence. Determine the ratio of the number of grid points and that of the computing time, if the reynolds number is increased by a factor of 2

The required number of grid points for DNS is proportional to Re ^9/4 (Rogallo & Moin 1984), where the grid spacings should be sufficiently fine to resolve the dissipation length 34 H. Choi and P. Moin scale. However, in this estimate, the Reynolds number is defined in terms of large eddy characteristic velocity and length scales. We derive the required number of grid points for DNS based on the streamwise length Lx, as was done for wall-modeled and wall-resolving LES. Let us consider a small computational box of dimensions dx × dy × dz inside the boundary layer. The Kolmogorov length scale is estimated from the dissipation rate in a turbulent boundary layer, , where the overline denotes time averaging. The dissipation rate is largest at the wall, i.e.,

at the wall, (2.7)

and decays rapidly away from the wall. Here, u and w are the streamwise and spanwise velocities, respectively, and the prime denotes the velocity fluctuations. Note that we include the mean velocity gradient term in Eq. (2.7) because the main concern of the present analysis is the grid resolution near the wall. Among the three terms on the right hand side of Eq. (2.7), the first term is largest: the ratios of second and third terms to the first one are much less than 1 (e.g., the ratio of second to first terms is about 0.16), but they increase weakly with the Reynolds number (Hu et al. 2006; Orl¨u & Schlatter ¨ 2011). Therefore, we take the first term in Eq. (2.7) and use Eq. (1.7) to estimate the Kolmogorov length scale in turbulent boundary layer. When uniformly spaced grids are used along dx, the number of grid points to resolve the Kolmogorov length scale is

Then, the total number of grid points required for DNS in this small computational box is

The total number of grid points for the entire computational domain is obtained by integrating Eq. (2.9) over the domain (Lx × δ × Lz): i.e.,

Using Eq. (1.6), we finally obtain

Discuss the main differences between turbulence modeling based on RANS and LES, in terms of modelling aspects and also of computational effort.

Otvet: In RANS averaging is performed (over time). By definition, RANS variables do not depend on time.

In LES averaging is performed locally over space (a small zone around each point) LES variables are time- dependent.

Modeling: RANS is using almost no laminar flame equation: everything is modeled

R(r)=

LES is computationally expensive at high Reynolds numbers.

Re=

(4) For what types of flows should an LES carried out instead of a simulation based on RANS. Give 3 examples with arguments for your answer.

ANSWER:

RANS can only give a time averaged mean value for velocity field. Since it is based on time averaging. In fact, velocity field in this method is averaged over a time period of "t" which is considerably higher than time constant of velocity fluctuations. Therefore, within the period of "t" we have only a constant mean velocity and could not monitor its time-dependent variations. For example, suppose that you can take successive pictures of velocity vector in a specific point of a turbulent flow during "t". Clearly, you can see that this velocity vector varies with the time. However, if you use RANS you can only see a constant velocity vector, which is an average of pictures you have already taken!
On the other hand, LES is based on filtering rather than averaging. In this method, you need to choose a filter size first. All flow scales larger than the filter size specified will be exactly calculated and the scales smaller than filter size will be modeled. Now consider the picture taking again. If you use LES, you can clearly see the variation of velocity vector at that specific point. Smaller the filter size, more exact the time variation resolution of velocity vector.

Limitation of RANS models:

Turbulence models based on RANS equations cannot predict unsteady turbulent fluctuations RANS model are often deliver inaccurate results for flow problems, in which strong separation, adverse pressure gradients or embedded vortices exist!

Large Eddy Simulation:

The larger scales in the turbulent flow are generated by sheared flow regions, which are mainly generated by the geometry and cannot easily be described in general Since the smaller scales become more and more isotropic, it should be easier to find a more accurate model for the small turbulent scales Goal: simulate the larger scales directly and model only the smaller scales In LES turbulent scales are part of the simulation, therefore a three dimensional unsteady simulation is required!

EXERCISE 3

(1)

EXERCISE 4

(1)

a) Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents,[1] and first explored by Deardorff (1970).[2] LES is currently applied in a wide variety of engineering applications, including combustion,[3] acoustics,[4] and simulations of the atmospheric boundary layer.[5] Large eddy simulation (LES) is a popular technique for simulating turbulent flows. The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged[1] 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. 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. For a stationary, incompressible Newtonian fluid.

RANS	LES
Without gradient For solution we can use 10 points Time average	Gradient are stronger Many points in the many time iteration Space filters

b) The Navier-Stokes equations are only valid as long as the representative physical length scale of the system is much larger than the mean free path of the molecules that make up the fluid. In that case, the fluid is referred to as a continuum. The ratio of the mean free path, λ, and the representative length scale, L, is called the Knudsen number, Kn=λ/L

c) LES: For incompressible flow, the continuity equation and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation, and the filtered Navier–Stokes equations, For the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered. This gives:

RANS:

(2) Sin(ax)sin(ax-b)dx=1/4(2axcos(b)+sin(b-2ax))

Sin(ax)sin(ax-b)dx=

1/2(cos(b)-cos(2ax-b))=

1/2cos(b)-1/2cos(2ax-b)=

1/2sin(b)+1/4asin(b-2ax)=

1/4a(2asin(b)+sin(b-2ax))

EXERCISE 5

(1) Explain the difference between a structured and unstructured mesh and discuss the advantages and disadvantages. Sketch an example for a structured and an unstructured mesh.

1) Struched mesh:

Advantages:

Computational efficiency;

Memory efficiency;

Higher-order scheme feasible monotonous

Disadvantages:

Monotonous tedious to construct;

Impractical for complex geometric

Unstructured mesh:

Advantages:

Geometric flexibility;

Suited well solution-based for adaptation;

Can be fitted to boundaries with feature;

Disadvantages:

Algorithm tend to be more complex;

Slower memory access;

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