Analysis of Turbulent (Infected by Disturbance) Flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Quantification of the Infection & its Effect on Mean Fow....

Averaging Navier Stokes equations for Study of Turbulent Flows Substitute into Steady incompressible Navier Stokes equations Instantaneous velocity fluctuation around average velocity Average velocity time Continuity equation:

Averaging of x-momentum Equation Write x-momentum equations in a short format:

Reynolds Averaged Steady Turbulent Momentum Equations Reynolds averaged x-momentum equation for steady incompressible turbulent flow

The Reynolds View of Cross Correlation

Reynolds averaged y-momentum equation for steady incompressible turbulent flow Reynolds averaged z-momentum equation for steady incompressible turbulent flow

Reynolds Averaged Navier Stokes equations

Reynolds Stress Tensor This is usually called the Reynolds stress tensor Reynolds stresses : total 9 - 6 are unknown Total 4 equations and 4 + 6 = 10 unknowns

Time averaged Infected Navier Stokes Equation For all the Three Momentum Equations, turbulent stress tensor:

Reynolds stresses Performing the Reynolds Averaging Process, new terms has arisen, namely the Reynolds-stress tensor: This brings us at the turbulent closure problem, the fact that we have more unknowns than equations. Three velocities + pressure + six Reynolds-stresses Three momentum equations + the continuity equation To close the problem, we need additional equations to solve infected flow.

Derivation of Conservation Equations for Reynolds Stresses Derivations of Reynolds-stress conservation Equations Introduces new unknowns (22 new unknowns)

Simplified Reynolds Averaged Navier Stokes equations 4 equations 5 unknowns → We need one more ???

Modeling of Turbulent Viscosity Fluid property – often called laminar viscosity Flow property – turbulent viscosity MVM: Mean velocity models TKEM: Turbulent kinetic energy equation models

MVM : Eddy-viscosity models Compute the Reynolds-stresses from explicit expressions of the mean strain rate and a eddy-viscosity, the Boussinesq eddy-viscosity approximation The k term is a normal stress and is typically treated together with the pressure term.

Algebraic MVM Prandtl was the first to present a working algebraic turbulence model that is applied to wakes, jets and boundary layer flows. The model is based on mixing length hypothesis deduced from experiments and is analogous, to some extent, to the mean free path in kinetic gas theory. Turbulent transport Molecular transport

Kinetic Theory of Gas The Average Speed of a Gas Molecule

Kinetic Theory of Gas Boundary Layer Motion of gas particles in a laminar boundary layer?

Microscopic Energy Balance for A Laminar BL Macro Kinetic Energy Random motion of gas molecules Solid bodies Dissipate this energy by friction Gas Molecules Dissipate this energy by viscosity at wall Thermal Energy Enthalpy = f(T) http://www.granular.org/granular_theory.html

Prandtl’s view of Viscosity For a gas in a state of thermodynamic equilibrium, the quantities such as mean speed, mean collision rate and mean free path of gas particles may be determined. Boltzmann explained through an equation how a gas medium can have small macroscopic gradients exist in either (bulk) velocity, temperature or composition. The solutions of Boltzman equation give the relation between the gradient and the corresponding flux in each case in terms of collision cross-sections. Coefficients of Viscosity, Thermal conductivity and Diffusion are thereby related to intermolecular potential.

Pradntl’s Hypothesis of Turbulent Flows In a laminar flow the random motion is at the molecular level only. Macro instruments cannot detect this randomness. Macro Engineering devices feel it as molecular viscosity. Turbulent flow is due to random movement of fluid parcels/bundles. Even Macro instruments detect this randomness. Macro Engineering devices feel it as enhanced viscosity….!

Prandtl Mixing Length Hypothesis The fluid particle A with the mass dm located at the position , y+lm and has the longitudinal velocity component U+U is fluctuating. This particle is moving downward with the lateral velocity v and the fluctuation momentum dIy=dmv. It arrives at the layer which has a lower velocity U. According to the Prandtl hypothesis, this macroscopic momentum exchange most likely gives rise to a positive fluctuation u >0. Y y X

Definition of Mixing Length Particles A & B experience a velocity difference which can be approximated as: The distance between the two layers lm is called mixing length. Since U has the same order of magnitude as u, Prandtl arrived at By virtue of the Prandtl hypothesis, the longitudinal fluctuation component u was brought about by the impact of the lateral component v , it seems reasonable to assume that

Prandtl Mixing Length Model Thus, the component of the Reynolds stress tensor becomes The turbulent shear stress component becomes This is the Prandtl mixing length hypothesis. Prandtl deduced that the eddy viscosity can be expressed as

Prandtl Mixing Length Model Thus, the component of the Reynolds stress tensor becomes The turbulent shear stress component becomes This is the Prandtl mixing length hypothesis. Prandtl deduced that the eddy viscosity can be expressed as

Fully Developed Duct Flow For x > Le, the velocity becomes purely axial and varies only with the lateral coordinates. V= W = 0 and U = U(y,z). The flow is then called fully developed flow. For fully developed flow, the Reynolds Averaged continuity and momentum equations for incompressible flow are simplified as: With

Turbulent Viscosity is a Flow Property The true Reynolds Averaged momentum equations for incompressible fully developed flow is:

Fully Developed Turbulent flow in a Circular Pipe: Modified Hagen-Poiseuille Flow The single variable is r. The equation reduces to an ODE: The solution of above Equation is: ????? Engineering Conditions: The velocity cannot be infinite at the centerline. Is this condition useful???

Estimation of Mixing Length To find an algebraic expression for the mixing length lm, several empirical correlations were suggested in literature. The mixing length lm does not have a universally valid character and changes from case to case. Therefore it is not appropriate for three-dimensional flow applications. However, it is successfully applied to boundary layer flow, fully developed duct flow and particularly to free turbulent flows. Prandtl and many others started with analysis of the two-dimensional boundary layer infected by disturbance. For wall flows, the main source of infection is wall. The wall roughness contains many cavities and troughs, which infect the flow and introduce disturbances.

Quantification of Infection by seeing the Effect Develop simple experimental test rigs. Measure wall shear stress. Define wall friction velocity using the wall shear stress by the relation Define non-dimensional boundary layer coordinates.

Approximation of velocity distribution for a fully turbulent 2D Boundary Layer

Approximation of velocity distribution for a fully turbulent 2D Boundary Layer For a fully developed turbulent flow, the constants are experimentally found to be =0.41 and C=5.0.

Measures for Mixing Length Outside the viscous sublayer marked as the logarithmic layer, the mixing length is approximated by a simple linear function. Accounting for viscous damping, the mixing length for the viscous sublayer is modeled by introducing a damping function D. As a result, the mixing length in viscous sublayer: The damping function D proposed by van Driest with the constant A+ = 26 for a boundary layer at zero-pressure gradient.

Based on experimental evaluation of a large number of velocity profiles, Kays and Moffat developed an empirical correlation for that accounts for different pressure gradients and boundary layer suction/blowing. For zero suction/blowing this correlation reduces to: With

Van Driest damping function

Distribution of Mixing length in near-wall region

Mixing length in lateral wall-direction

Conclusions on Algebraic Models Few other algebraic models are: Cebeci-Smith Model Baldwin-Lomax Algebraic Model Mahendra R. Doshl And William N. Gill (2004) Gives good results for simple flows, flat plate, jets and simple shear layers Typically the algebraic models are fast and robust Needs to be calibrated for each flow type, they are not very general They are not well suited for computing flow separation Typically they need information about boundary layer properties, and are difficult to incorporate in modern flow solvers.

Steady Turbulent flow

A Segment of Reconstructed Turbulent Flame in SI Engines

Large Scales: Parents Vortices

Creation of Large Eddies an I.C. Engines There are two types of structural turbulence that are recognizable in an engine; tumbling and swirl. Both are created during the intake stroke. Tumble is defined as the in-cylinder flow that is rotating around an axis perpendicular with the cylinder axis. Swirl is defined as the charge that rotates concentrically about the axis of the cylinder.

Instantaneous Energy Cascade in Turbulent Boundary Layer. A state of universal equilibrium is reached when the rate of energy received from larger eddies is nearly equal to the rate of energy of when the smallest eddies dissipate into heat.

One-Equation Model by Prandtl A one-equation model is an enhanced version of the algebraic models. This model utilizes one turbulent transport equation originally developed by Prandtl. Based on purely dimensional arguments, Prandtl proposed a relationship between the dissipation and the kinetic energy that reads where the turbulence length scale lt is set proportional to the mixing length, lm, the boundary layer thickness  or a wake or a jet width. The velocity scale is set proportional to the turbulent kinetic energy as suggested independently. Thus, the expression for the turbulent viscosity becomes: with the constant C to be determined from the experiment.

Transport equation for turbulent kinetic Energy x-momentum equation for incompressible steady turbulent flow: Reynolds averaged x-momentum equation for incompressible steady turbulent flow: subtract the second equation from the second equation to get Multiply above equation with u and take Reynolds averaging

Similarly: Define turbulent kinetic energy as:

Turbulent Kinetic Energy Conservation Equation The Cartesian index notation is: Boundary conditions:

Taylor Hypothesis Taylor proposed an hypothesis that the energy transport contribution of small size eddies that are carried by a large scale eddy Taylor proposed an hypothesis: The energy transport contribution of small size eddies that are carried by a large scale eddy, compared with the one produced by a larger eddy, is negligibly small. In such a situation, the transport of a turbulence field past a fixed point is due to the larger energy containing eddies. It states that “in certain circumstances, turbulence can be considered as “frozen” as it passes by a sensor”.

Spectral Representation of Turbulent Flows

Kolmogorov Hypotheses Kolmogorov established his universal equilibrium theory based on two similarity hypotheses for turbulent flows. The first hypothesis states that for a high Reynolds number turbulent flow, the small-scale turbulent motions are isotropic and independent of the detailed structure of large scale eddies. Furthermore, there is a range of high wavenumbers where the turbulence is statistically in equilibrium and uniquely determined by the energy dissipation and the kinematic viscosity . With this hypothesis and in conjunction with dimensional reasoning, Kolmogorov arrived at length (), time () and the velocity (v) scales. Considering the Kolmogorov’s length and velocity scales, the corresponding Kolmogorov’s equilibrium Reynolds number is

Structure of Equilibrium Turbulence

Structure of Equilibrium Turbulence

One and Two Equation Turbulence Models The derivation is again based on the Boussinesq approximation The mixing velocity is determined by the turbulent turbulent kinetic energy The length scale is determined from another transport equation ex.

Second equation

Dissipation of turbulent kinetic energy The equation is derived by the following operation on the Navier-Stokes equation The resulting equation have the following form

The k-ε model Eddy viscosity Transport equation for turbulent kinetic energy Transport equation for dissipation of turbulent kinetic energy Constants for the model

Dealing with Infected flows The RANS equations are derived by an averaging or filtering process from the Navier-Stokes equations. The ’averaging’ process results in more unknown that equations, the turbulent closure problem Additional equations are derived by performing operation on the Navier-Stokes equations Non of the model are complete, all model needs some kind of modeling. Special care may be need when integrating the model all the way to the wall, low-Reynolds number models and wall damping terms.