Simulations for Enhancing Aerodynamic Designs
2. Governing Equations and Turbulence Models
by
Dr. KANNAN B T, M.E (Aero), M.B.A (Airline & Airport), PhD (Aerospace Engg), Grad.Ae.S.I, M.I.E, M.I.A.Eng, M.I.A.CS.IT,
Conservation Laws
➢ Conservation of mass “Rate of increase of mass in a fluid element equals net rate of flow of mass into a fluid element”
➢ Conservation of Momentum “Rate of increase of momentum of fluid particle equals sum of forces acting on fluid particle”
➢ Conservation of Energy “Rate of increase of energy of fluid particle equals sum of net rate of heat added to fluid particle and net rate of work done on fluid particle”
✔ Scalar Transport It deals with the transport of active and passive scalars.
Governing Equations for Turbulent Flows
Instantaneous equations ↓ Reynolds Decomposition Decomposed equations ↓ Reynolds Averaging Reynolds Averaged Equations
Reynolds Decomposition
Instantaneous velocity = Mean velocity + Fluctuating velocity
Averaging
RANS
Navier Stokes Equations subjected to Reynolds Decomposition and Reynolds Averaging yields Reynolds Averaged Navier Stokes Equations and it is usually called as RANS.
Instantaneous Governing Equations
Reynolds Mean Momentum Equation
RANS CLOSURE
RANS = Mean momentum equation having Reynolds Stress / Apparent Mean Stress as additional term which was generated by the averaging process.
Hence, Number of unknowns > Number of equations ↓ CLOSURE PROBLEM OF TURBULENCE
Solution: Modeling for the unknown
New Unknown
The closure problem is due to Reynolds Stress resulting from the non-linear term from Navier Stokes equations. This non-linear term is responsible for the energy cascading.
Energy transfer takes place by “ENERGY CASCADE”
Reynolds Stress
The Reynolds Stress (RS) is defined by the Tensor
Equation for Reynolds Stress
The transport equation for RS in descriptive form Advection(A) + Transport(T) = Production(P) - Dissipation(D)
Homogeneous flows (P-D=0) - average as invariant in any spatial direction - no gradients and no transports (grid turbulence).
Equations for Turbulent Flow Predictions
RANS – Reynolds Averaged Navier Stokes
Process
– Reynolds Decomposition
u(t) = U + u'
– Reynolds Averaging
RANS
Mean momentum equation
– U,V,W,P & Rij
– Rij (Reynolds Stress Tensor)
5 Terms & 4 Equations-->Closure Problem
Solution: Modeling
Turbulence Modelling
● Eddy Viscosity Hypothesis:
– is similar to Newtons law of viscosity
● Bossinesq Approximation:
– Reynolds stress might be proportional to mean
rates of deformation
Scales of Turbulence
● Velocity scale
● Length Scale
● Time Scale
Any two is enough to describe a turbulent flow
MODELLING TECHNIQUES
● RANS
– Reynolds Averaged Navier Stokes Equation
● LES
– Large Eddy Simulation
● DES
– Detached Eddy Simulations
DNS
– Direct numerical Simulations
Classification of Models
● Algebraic Equations or Zero Equation ● One Equation(Spalart Allmaras)
● Two Equation(k-epsilon, k-omega)
● Two Equation – Derived (RNG, SST etc..)
● Three Equation (k-kl-omega) ● Four Equation (k-epsilon-Q-Zeta)
● Higher order (RST)
SPALART - ALLMARAS
➢ Also a simplest model ➢ It has only one transport equation for kinematic eddy viscosity.
➢ The model constants are from experiments carried out on airfoils using wind tunnels
➢ Heavily tuned for external aerodynamics.
Two Equation Models
● Two equation models are better than zero and one equation models in a broader sense. ● They have separate transport equations for – length scale and – velocity scale.
Standard k-ω model
● This model was developed by Wilcox ● It is similar to the standard k-ε model
● The dissipation is replaced by turbulent frequency.
● It has two transport equations – Turbulent Kinetic Energy (k) and – Turbulence Frequency (ω).
SST k-ω model
● This model was developed by Menter ● It is similar to the standard k-ω model ● The omega equation is formed by substituting ε = kω. ● It has two transport equations – Turbulent Kinetic Energy (k) and – Turbulence Frequency (ω).
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