0112-湍流模型介绍

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Outline
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Turbulence models introduction Boussinesq hypothesis Eddy viscosity concept Zero equation model One equation model Two equation models Algebraic stress model Reyolds stress model Comparison Applications Developments Conclusion
ij
T
U i U j ui ' u j ' t x xi j






A new quantity appears: the turbulent viscosity or eddy viscosity (νt ). The second term is added to make it applicable to normal turbulent stress. The turbulent viscosity depends on the flow, i.e. the state of turbulence. The turbulent viscosity is not homogeneous, i.e. it varies in space.
Turbulence Models and Their Applications
Turbulence models
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A turbulence model is a procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it:
Boussinesq hypothesis
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Many turbulence models are based upon the Boussinesq hypothesis.

It was experimentally observed that turbulence decays unless there is shear in isothermal incompressible flows. Turbulence was found to increase as the mean rate of deformation increases. Boussinesq proposed in 1877 that the Reynolds stresses could be linked to the mean rate of deformation.

Developed pipe flows
lm y y 0.14 0.08(1 ) 2 0.06(1 ) 4 R R R
R = radius of the pipe or the half width of the duct
Turbulence Models and Their Applications



Zero Equation Model - Mixing Length Model
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On dimensional grounds one can express the kinematic turbulent viscosity as the product of a velocity scale and a length scale:
Major Drawbacks
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Interaction among eddies is not elastic as in the case for molecular interactions in kinetic theory of gases. For many turbulent flows, the length and time scale of characteristic eddies is not small compared with the flow domain (boundary dominated flows). The eddy viscosity is a scalar quantity which may not be true for simple turbulent shear flows. It also fails to distinguish between plane shear, plane strain and rotating plane shear flows. Successful – 2D shear flows. Erroneous results for simple shear flows such as wall jets and channel flows with varying wall roughness.

Zero equation model: mixing length model. One equation model Two equation models: k- style models (standard, RNG, realizable), k- model, and ASM. Seven equation model: Reynolds stress model.

t uclc (or ) lc2 / tc

This concept assumes that Reynolds stress tensor can be characterized by a single length and time scales.
Turbulence Models and Their Applications
Turbulence Models and Their Applications

Equations for mixing length
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Wall boundary layers
lm Ky lm
(y / / K) (y / / K)
δ = boundary layer thickness y = distance from the wall λ = 0.09 K = Von-Karman Constant

Similarly, link Reynolds stresses to the mean rate of deformation
Turbulence Models and Their Applications
Eddy Viscosity Concept
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One of the most widely used concept Reynold’s stress tensor –
TURBULENCE MODELS AND THEIR APPLICATIONS
Presented by: T.S.D.Karthik Department of Mechanical Engineering IIT Madras Guide: Prof. Franz Durst
10th Indo German Winter Academy 2011
t (m 2 / s) (m / s) (m)

If we then assume that the velocity scale is proportional to the length scale and the gradients in the velocity (shear rate, which has dimension 1/s): U y we can derive Prandtl’s (1925) mixing length model: U t 2 m y Algebraic expressions exist for the mixing length for simple 2-D flows, such as pipe and channel flow.
Turbulence Models and Their Applications
Eddy Viscosity Concept
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It is, however, assumed to be isotropic. It is the same in all directions. This assumption is valid for many flows, but not for all (e.g. flows with strong separation or swirl). The turbulent viscosity may be expressed as
Turbulence Models and Their Applications


Classification