AOE 6154 TURBULENT SHEAR FLOW01/22/09 VERSION
GLOSSARY OF TERMS AND CONCEPTS
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NATURE OF FLUID TURBULENCE:
TURBULENT FLOW is an UNSTEADY FLOW with THREEDIMENSIONAL COHERENT VORTICAL STRUCTURES of VARIOUS SIZES AND VELOCITIES that have
LIMITED LIFETIMES AND SPATIAL EXTENT. For the same initial and boundary conditions, the same mean flow statistics occur, i.e. the MEAN FLOW IS STATISTICALLY ERGODIC.
*BACKFLOW (see SEPARATION)
*BURSTING
The rapid chaotic diffusion of fluid after it has been ejected (EJECTION) in a coherent motion.
Bursting in wall flow (Pope, Turbulent flows)
With increasing downstream distance, a low speed streak migrates slowly away from the wall; and then, at some point (typically around y+10), it turns and moves away from the wall more rapidly – a process referred to as streak lifting, or ejection. As it is lifted, the streak exhibits a rapid oscillation followed by a breakdown into finer-scale motions.
CASCADE OF TURBULENT KINETIC ENERGY
The process of energy transfer from large scale coherent structures to small scale coherent structures. This process is driven by vortex stretching and ultimately leads to viscous dissipation of the smallest scale structures.
CELERITY (p. 2-6, RLS)
The wave speed of vortical motions (eddies);the speed of a given size turbulent coherent motion to move from one spatial location to another:
d( )/dt = (Uc)(d( _)/dx).
TAYLOR’S HYPOTHESIS states that this is the local mean velocity. Experimental data show that the celerity approaches the local mean velocity only for the small-scale motions. The celerity values for low frequencies (large-scale motion) are generally different from the local mean due to induced velocities from neighboring large eddies.
*CHAOS and CHAOTIC MOTION
Chaotic behavior is common in motions governed by time-dependent non-linear equations, such as the Navier-Stokes equations. For some domain of variables, non-unique and therefore unpredictable time-dependent solutions of the Navier-Stokes equations are produced with infinitesimal changes in initial conditions.
CLOSURE (p. 87, P)
The introduction of modeling equations in order to provide as many equations as unknowns.
Reynolds-averaged equations contain terms, such as Reynolds stresses, for which model equations are needed.
From (Pope, Turbulent flows)
For a general statistically 3D flow, there are four independent equations governing the mean velocity field; namely three components of the Reynolds equations together with either the mean continuity equation or the Poisson equation for p. However, these four equations contain more than four unknowns. This is a manifestation of the closure problem. They cannot be solved unless the Reynolds stresses are somehow determined.
COHERENCY
The coherency 0 γ2 (f) 1 for a given frequency f is defined mathematically as the complex cross-spectrum of 2 time-dependent signals times the complex conjugate cross-spectrum, divided by the auto-spectra of each of the 2 signals. A coherency of unity means that the 2 signals at frequency f look exactly alike, even if they are phase-shifted.
COHERENT
Two signals or measures of a time-dependent phenomenon are called coherent if they look alike in some way. Example: velocity fluctuation signals at two spatial locations within a fluid structure will look alike over some frequency range or bandwidth. The COHERENCY from these 2 signals will have values well above zero.
*COHERENT MOTION (or TURBULENT STRUCTURE) (p. 2 of Robinson)
“A three-dimensional region of the flow over which at least one fundamental flow variable (velocity component, density, temperature, etc.) exhibits significant correlation with itself or with another variable, over a range of space and/or time that is significantly larger than the smallest local scales of the flow.”
*COHERENT MOTION CLASSIFICATION BY QUADRANTS (ideas from Robinson, p. 2)
In u versus v co-ordinates, SWEEPS of higher velocity fluid toward lower-velocity regions are fourth quadrant u > 0, v < 0 motions; EJECTIONS of low-speed fluid from low-velocity regions are second quadrant u < 0, v > 0 motions; INTERACTIONS are first quadrant u > 0, v > 0 motions and third quadrant u < 0, v < 0 motions. SWEEPS and EJECTIONS produce positive Reynolds shearing stresses while INTERACTIONS produce negative Reynolds shearing stresses. Note that for mean two-dimensional flows, there are equal contribution for w > 0 and w< 0.
For mean three-dimensional flows, in general contributions for w> 0 and w < 0 are different, so one can classify all motions in terms of the OCTANTS: sweeps, u > 0, v < 0, for w > 0 and w< 0 ; ejections, u< 0, v > 0, for w > 0, and w< 0; interactions, u> 0, v > 0 and u< 0, v< 0, for both w > 0, and w< 0.
COHERENT TURBULENCE (p. 309, Hussain, 1986)
The time-dependent coherent structure that was EDUCTED from signals within a turbulent flow. INCOHERENT TURBULENCE is the difference between the total turbulent fluctuation signal and the educted coherent signal at that location. Turbulence which appears incoherent over large length scales may be coherent over smaller length scales.
COHERENT VORTICAL STRUCTURE (EDDY) (Ideas from Hussain, 1986)
A limited spatial region of a flow that contains phase-correlated vorticity and identifiable structure that is distinct from other regions in the flow. Within the length and time scales of this flow region over a short time signal record, there exist values of the COHERENCY well above zero.
*COUETTE FLOW
Flow between 2 parallel plates, with one wall moving with a velocity and the second at rest.
CROSS SPECTRUM (Bendat and Piersol)
The cross-spectrum Gxy(f) = Cxy(f) - iQxy(f) at frequency f for time varying signals x(t) and y(t). The CO-SPECTRUM Cxy(f) is obtained by multiplying the narrow-bandpass Be portions of x(t) and y(t) at frequency f and time-averaging over time T. The QUADRATURE Qxy(f) at frequency f is obtained by multiplying the narrow-bandpass Be portions of x(t) and a 90 degree phase shift of y(t) and time-averaging over time T. The phase shift φ(f) between the 2 spectral functions at frequency f is given by φ(f) = tan-1 (Qxy(f)/ Cxy(f)).
*DETACHMENT (D) (see SEPARATION) ( p. 460, Simpson)
Location along a mean two-dimensional flow where there is on the average 50% of the time BACKFLOW or upstream moving flow. Also location where the mean wall shearing stress is zero.
DIFFUSION
A process of mixing of fluid particles that causes a fluid property at a point in a flow to spread, or be transferred away from that point.
DIMENSIONAL ANALYSIS (taken from TL p.5 modified)
The dimensional units of a small number of independent variables or parameters can dictate the functional form of a relation with a dependent variable; numerical constants are determined by theoretical or experimental results. An example of this is the form of the “inertial subrange” of the turbulent kinetic energy spectrum.
DISSIPATION OF TURBULENCE KINETIC ENERGY (p. 3, TL)
A process in which shear stresses perform deformation work that increases the internal energy of the fluid at the expense of kinetic energy of the turbulence.
(Mathieu & Scott, An introduction to turbulent flow)
Turbulent kinetic energy of mean flow by mean flow instabilities turbulent kinetic energy of large scales energy flux through small scales dissipation by viscosity or shear stresses.
*DYNAMIC SUBLAYER (see VISCOUS SUBLAYER)
EDDY (see COHERENT VORTICAL STRUCTURE)
EDUCTION ( p. 309, Hussain, 1986)
The process of measuring the properties of a coherent vortical structure over its spatial extent. It implies ensemble-averaging over a large number of successive coherent structures with similar time-dependent signal signatures or “preferred modes”.
*EJECTIONS (see COHERENT MOTION CLASSIFICATION BY QUADRANTS)
Ensemble average (Pope, Turbulent flows)
The ensemble average (over N repetitions) is defined by
ENSTROPHY
The root-mean square of the vorticity fluctuation.
*ENTRAINMENT OF NONTURBULENT FLUID
The process of converting nonturbulent fluid to turbulent fluid. The boundary between turbulent fluid and nonturbulent fluid (VISCOUS SUPERLAYER) has an irregular, timedependent shape with a very large interfacial area. Sequentially, instabilities in the flow cause indentations in this boundary to appear, grow to large amplitudes, and ERUPT (or rapidly move) into the surrounding non-turbulent fluid. The erupting turbulent fluid (three-dimensional bulges that are of the size or scale of the shear layer thickness) then rolls up and ENGULFS (or surrounds) some of the nonturbulent fluid. The engulfed nonturbulent fluid is then convected with the shear layer. Viscous mixing (or “SMALL-SCALE NIBBLING” at Kolmogorov scales) occurs at the viscous superlayer, transmitting vorticity to the engulfed fluid and making it turbulent. (Also see Hussain, p. 331)
*ENTRAINMENT VELOCITY, VE (p. 4-24, Notes, Simpson)
The entrainment volumetric flux (volume per unit time/unit area) of engulfed non-turbulent fluid across a surface perpendicular to the mainstream flow direction.
*EFFECTIVE ENTRAINMENT VELOCITY, VE (problem 4.8, Notes, Simpson )
The volumetric flux (volume per unit time/ unit area) of free-stream fluid that enters the shear layer at the δ0.99 location. Note that it is given by VE = {0.99Umax + 0.01Umin}d(δ0.99)/dx - V at δ0.99 .
ERGODIC (see STATISTICALLY ERGODIC)
EQUILIBRIUM FLOWS (SELFSIMILAR FLOWS)
Flows in which ratios of flow properties to one another can be formed that remain invariant in some spatial coordinate. This property can be used to reduce the partial differential equation (PDE) describing the flow to an ordinary differential equation (ODE) that is described in terms of a similarity spatial variable. However, all boundary conditions (BCs) and initial conditions (ICs) of the PDE must still be satisfied. Since the newly formed ODE will not be able to enforce arbitrary BC and ICs of the PDE, some of the BCs and ICs must be equivalent in terms of the similarity spatial variable and be satisfied simultaneously.
*EULER FLOW
A flow described by the inviscid Euler equations, with or without vorticity. Usually describes cases with vorticity since cases without vorticity are termed potential flows.
*FULLY DEVELOPED FLOW
Used to specify the condition ∂( )/ ∂x = 0 in fully-constrained constant cross-sectional area and shaped flow channels. Here x is the direction along the flow.
*HELICITY and HELICITY DENSITY (p. 335, Hussain)
Overall global transported vorticity by velocity vector.H = ∫u●ω dV = ∫hdV where H is the HELICITY, which is a scalar, h is the HELICITY DENSITY, u is the VELOCITY VECTOR, and ω is the VORTICITY VECTOR, and V is the volume of the domain. Note that |u x ω|2 + |u ● ω|2 = |u| 2 |ω|2 from a trigonometric identity.
HELMHOLTZ'S VORTEX THEOREMS ( Truesdell, 1954)
I. The strength of a vortex tube is constant along its length. The circulation around a crosssection of a vortex tube is equal to its strength. (In the form given by Kelvin: Circulations taken in the same sense about any two reconcilable circuits lying on the surface of a vortex tube are the same.) The average magnitude of the vorticity inside a vortex tube is inversely proportional to the crosssectional area of the vortex tube.
II. In the absence of non-conservative body forces and viscous torques in a barotropic fluid (density is only a function of pressure), vorticity is convected with the fluid and does not diffuse to other pieces of mass. VORTEX LINES ARE MATERIAL LINES IN AN INVISCID FLUID. This follows from KELVIN'S THEOREM which states that under these conditions the substantial derivative of circulation is zero.
III. In a fluid motion in which the vortex lines are material lines, the necessary and sufficient condition for the strength of the vortex tubes to remain unchanged with time is that the motion be circulation-preserving.
*INCIPIENT DETACHMENT (ID) ( p. 460, Simpson)
Location along a mean two-dimensional flow where there is on the average 1% of the time backflow or upstream moving flow.
INSTABILITY OF MEAN FLOW
The condition in which small perturbations of the mean flow are amplified until they ultimately form coherent vortical structures. This condition causes a laminar flow to transition to turbulent flow.
*INTERACTIONS (see COHERENT MOTION CLASSIFICATION BY QUADRANTS)
*INTERMITTENCY, γ
Long-time-averaged fraction of time at a given location in a flow when the flow is turbulent. γ = ∫β(t)dt/T, where the integral is over time T and β(t) is the INSTANTANEOUS INTERMITTENCY which has a value of unity (1) for the presence of turbulent fluid and zero for non-turbulent fluid.
*INTERMITTENT TRANSITORY DETACHMENT (ITD) ( p. 460, Simpson)
Location along a mean two-dimensional flow where there is on the average 20% of the time backflow or upstream moving flow.
(ASYMPTOTIC) INVARIANCE (ideas from TL pp. 56)
Turbulent flows are characterized by very high Reynolds numbers; as the Reynolds number approaches infinity, any proposed descriptions of turbulence should not be influenced by viscosity except at the smallest scales. Asymptotic invariance requires Reynolds number similarity.
(LOCAL) INVARIANCE (SELF PRESERVATION) (p. 6, TL)
A flow has this property when flow characteristics at some point in time and space are controlled mainly by the immediate environment. The turbulence is dynamically similar everywhere when non-dimensionalized with local length and time scales.
ISOTROPIC TURBULENCE
The condition in which averages of turbulence quantities are invariant under coordinate system rotation or reflection. In real flows this property is restricted to the smallest scales of turbulence at high Reynolds number. The small eddies are not correlated with the large eddies. Isotropy at small scales is called LOCAL ISOTROPY.
RETURN TO ISOTROPY
This refers to the tendency of turbulence of all scales and origins to decay to an isotropic condition at small scales.
Kolmogorov’s hypothesis of local isotropy (Pope, Turbulent flows)
At sufficiently high Reynolds number, the small scale turbulent motions are statistically isotropic.
KOLMOGOROV'S FIRST SIMILARITY HYPOTHESIS (from RLS p. 218)
The mean properties of the small eddies of any large Reynolds number motion are uniquely determined by the kinematic viscosity and the average rate of energy dissipation per unit mass.
KOLMOGOROV'S SECOND HYPOTHESIS (from RLS p. 218)
For sufficiently large Reynolds numbers, there is a subrange of small eddies in which its average properties are determined only by the average rate of dissipation per unit mass. This subrange is called the INERTIAL SUBRANGE and eddies within this range are large enough
that viscosity does not have an appreciable effect.
(Pope, Turbulent flows)
In every turbulent flow at sufficiently high Reynolds number, the statistics of the motions of scale l in the range l0 > l > have a universal form that is uniquely determined by , independent of .
*LEAP-FROGGING VORTICES (see PAIRING OF LARGESCALE COHERENT STRUCTURES)
LOCAL (ENERGY) EQUILIBRIUM
The situation in which the energy transferred into some wavenumber range is equal to the energy transferred out. The convective transport terms in the governing equations are negligible and there is a balance between incoming TKE and the dissipation and diffusion processes. The ratio of terms in the transport equations at each wavenumber is the same. LOCAL EQUILIBRIUM exists within the INERTIAL SUBRANGE.
*MIXING LAYER
The flow between two flow regions with different flow velocities and/or different scalars such as temperature and mass species.
PAIRING OF LARGESCALE COHERENT STRUCTURES
The high Reynolds number large-scale vortical structures within a shear layer behave qualitatively like inviscid flow. Therefore groups of large scale vortical structures interact and induce velocities on one another as described by the Law of BiotSavart. This causes pairs of large scale vortical structures to rotate about one another (LEAP FROG), while vorticity is diffused outward causing the pair to agglomerate, or merge together into one larger vortical structure.
POCKET ( SEE Robinson)
RAPID DISTORTION THEORY (pp. 404 - 406, Pope)
The distortion of the flow dominates the changes, with Reynolds shearing stresses and viscous effects causing negligible effect. In essence, the instantaneous flow is governed by time-dependent Euler’s equations for a vortical flow.
REALIZABILITY CONDITIONS FOR REYNOLDS STRESSES (Schumann)
There are physical bounds on the values that Reynolds stresses can obtain within a flow. While actual experimental flows do not violate these realizability conditions, some faulty experimental data sets and calculations using some turbulence models have been known to violate these conditions. Condition #1: Each of the time-averaged Reynolds normal stresses, u2, v2, and w2 must be positive definite at all locations and in all possible co-ordinate systems. The sum of these three normal stresses, which is twice the specific turbulent kinetic energy, is also positive definite and is independent of the co-ordinate system. Since we can find a principal axes co-ordinate system for any stress tensor, this also means that the determinant of the stress tensor must also be positive. Condition #2: The magnitudes of the Reynolds-averaged shearing stresses |-uv|, |-uw|, and |-vw| must satisfy |-uv|2≤ (u2)(v2), |-uw|2≤ (u2)(w2), and |-vw|2≤ (v2)(w2), otherwise the cross-correlation between two different fluctuations would be greater than the correlation of a fluctuation with itself.
*REATTACHMENT (R) (see SEPARATION) ( p. 460, Simpson)
Location downstream of separation along a mean two-dimensional flow where there is on the average 50% of the time BACKFLOW or upstream moving flow. Also location where the mean wall shearing stress is zero.
REYNOLDS NUMBER
Is calculated as (velocity scale)*(length scale) / (kinematic viscosity). It is the ratio of the inertial forces of a flow or turbulence of a given scale to the viscous forces acting on the flow.
REYNOLDS DECOMPOSITION
Variables are separated into a time-mean part and a time-fluctuating part. The mean part is a longtime average value of the flow variable. The fluctuating part is the remainder. By definition the mean value of the fluctuating part is zero.
REYNOLDS AVERAGING
Use Reynolds decomposition of each quantity in an equation and then take a longtime
average. Since the fluctuating part of each variable has a zero mean value, the fluctuating part of LINEAR terms drops out of the equation. The products of fluctuation terms that arise from NONLINEAR terms in the governing fluid dynamic equations lead to non-zero terms that must be modeled to complete closure.
*SEPARATION (p.459 Simpson)
The entire process of ‘departure’ or ‘breakaway’ or the breakdown of boundary-layer flow. Thickening of the rotational flow next to the wall and significant values of the normal-to-wall velocity component separation. In mean two-dimensional flows, downstream of DETACHMENT there is a mean backflow or flow in the upstream direction. In mean three-dimensional flows, there is rarely any mean backflow.