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4. Generic Turbulent Flows and Convective Phenomena

• Classification of turbulent flows• Generic flows: features and relevance to modelling• Turbulent wall boundary layers

MAŠINSKI FAKULTET U SARAJEVU

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• Most flows of practical interest are in complex geometries.

Flow Classification – Canonical Turbulent Flows

• Homogeneous turbulent flows

• Thin shear flows (parabolic, boundary-layer flows):

- Free thin shear flows (jets, mixing layers, far-wakes)

- Wall boundary layers

• Impinging (stagnation) flow regions

• Separating flow regions

• Recirculating flows

• Swirling flows

• Flows with system rotation

• Flows dominated by thermal and/or concentration buoyancy

• …..

• For studying physics, validating models and solution methods, its is instructive toconsider simpler, “generic ” or “canonical ” flows dominated by only some phenomena (permits to neglect some terms in governing equations):

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Homogeneous Turbulent Flows

• Axisymmetric strain and return to isotropy

2 3 1

2 3 1

1

1

1

20

U U U

x x x

Ux

∂ ∂ ∂= = − ∂ ∂ ∂

∂ >∂

• Plane strain2 3

12 3

;U U

U constx x

∂ ∂= = −∂ ∂

• Simple (plane) shear1

2

Uconst

x

∂ =∂

• Decay of isotropic grid turbulence

;k

U const Ux

ε∂= = −∂

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Thin FreeFree Shear Flows

• Free (away from a solid wall) turbulent shear flows fall into category of thin shear flows (Navier-Stokes equations parabolized by Prandtlboundary layer approximations): 1 1

1 22 1

; ;U U

U U Lx x

δ∂ ∂>> >> <<∂ ∂

Jet Far wake Mixinglayer

Boundary layer

Wall Jet

• Typical thin free shear flows are plane and round jets, far wakes and mixing layers (shown here in parallel with two thin wall shear layers: flat plateboundary layer and wall jet).

x1 x1

x2

U1

δδδδL

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Thin WallWall Shear Flows

• Turbulent thin shear flows along solid walls are the classic wall boundary layers(internal or external, wall attached or separating, 2 or 3-dimensional.

• Solid wall is a strong source of vorticity and an indirect source of turbulence .

External boundary layerover an airfoil

3-D boundary layers (in corners)

(e.g. rotor-blade-and wing-bodyjunctions)

Flat-plate 2-D boundary layerInternal boundary layers (in channels and pipes)

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Some “more complex” generic flows

“Back-step” flow

Buoyancy-driven flows

Curved-wall separation, vortex shedding

Thot

Tcold

Impinging jets Externally driven cavity flows

Sharp-edge separation, recirculation, reattachment

Flows past bluff bodies

Stagnation, radial spreading

Enclosed recirculation)

Plumes, thermals,convective cells

Swirl and rotation

Rotational effect

D L

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• In many complex situations a reasonable insight into the problem can be gained by decomposing the flow domain into zones which correspond to some of the generic flow classes.

Complex flows decomposition

• Such an a priori analysis is also useful before deciding on the appropriate levelof mathematical model and computational code to be applied for obtaining the solution.

A – Accelerating BLD – Decelerating BLFW – Far wakeI - ImpingementNW – Near wake

RO - Rotation RA – ReattachmentC – RecirculationS – SeparationTS - Transverse shear

II

AA DD

A

SS SSS

RARA

SNW

FWRO

RARA

RO

DD

C

TS TS

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Wall-boundary layers

• Wall boundary layers are important because many flows of practical relevance are bounded in part or as a whole by solid walls.

• Between the viscous sublayer and the fully turbulent layer there is a transitional 'buffer' zone where both the viscous and inertial forces are of importance.

• The significance of boundary layers is especially relevant to deriving approximate methods for treating wall boundary conditions, which are essential in predicting accurately wall-bounded flows, heat and mass transfer.

• The simplest form is a boundary layer over a flat plate at zero pressure gradient, which can be analyzed using simple similarity arguments.

• At high Reynolds numbers, after a short transition length, initially laminar flow becomes fully turbulent and independent of the fluid viscosity

• However, a solid wall suppresses the velocity and its fluctuation ('no-slip conditions'), so that, irrespective of the Re number, there is always a thin

viscous sublayer attached to the wall.

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y

x

Non-turbulent fluid

δ(x)∼x4/5

Viscous sublayerInner wall region

Outer region

(wake)

laminar transition turbulent

U∞

δ(x)∼x1/2

Lδδδδ

Turbulent wall boundary layers

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Turbulent wall boundary layers: similarity analysis

U yf = U L

( , , , ,..)wU f yτ ρ µ=

2 22

2 2w w

w

y yUf f

ρ τ τρτ µ ν ρ

= =

• Wall boundary layers possess a self-similarity, so that normalised velocity (and other properties) can be expressed as:

where U and L are the characteristic velocity and length scales respectively

• For a constant-pressure flat-plate boundary layer, similarity arguments lead to:

• Dimensional analysis gives (assume: )

from which the characteristic velocity and length (“inner-wall”) scales are deduced:

w Uττρ

=U = = “friction velocity”

w Uτ

ρ νντ

=L = = “viscous length”

, , ,wU C yα β γ δτ ρ µ=

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Turbulent wall boundary layers: similarity analysis

• A general non-dimensional similarity expression for fluid velocity can be written now as:

or ( )yUU

f U f yU

τ

τ ν+ + = =

• It remains now to determine the function “f” !

• However, because of a multi-layer structure of a wall boundary layer, we need to consider separately the viscous and the fully turbulent zones

• In the wall-adjacent viscous sublayer, the viscous force (represented by µ) is dominant over the inertial forces (represented by fluid density ρ);

• In the fully turbulent layer sufficiently away from a solid wall, it is opposite:inertial forces prevail over viscous forces;

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Turbulent wall boundary layers (WBL): mechanical an alogy

• A mechanical system consisting of a spiral and a plate spring with stiffness C1

and C2 respectively, illustrate the two-layer composition of in a turbulent WBL

• Subjected to a constant force F, a change

in C1 to C’1 (representing fluid viscosity µ) affects only the position of the plate spring

(“velocity magnitude”) but not its shape(elastic line) which is governed by stiffness

C2 (representing turbulent viscosity)

• Thus, in the viscous sublayer velocity U and

dU/dy depend only on µ (and τw), whereas

Viscous sublayerBuffer layer

Turbulent layer

U

y

• In the turbulent zone, dU/dy is independent of µ!

• There is, of course a thin buffer zone in between

the two layers

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Turbulent wall boundary layers (WBL):

Viscous sublayer

• The only form of f that ensures that U is independent of ρ is f=1, leading to

U y+ +=which means that the velocity varies linearly with a distance from the wall!

Turbulent wall zone (turbulent inner wall layer)

• Here we expect that the shape of the velocity profile is independent of µ by

postulating that for U=U τ f(y+)

( )dUf

dyµ≠

• Thus, df/dy+ must be proportional to ν and nondimensional!

2UdU df dy dfU

dy dydy dyτ

τ ν

+

+ += =

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Turbulent wall boundary layers (TWBL)

1 1ln ln( )U y B Ey

κ κ+ + += + =

• The only form of df/dy+ that satisfies these conditions (see previous slide) is:

Turbulent wall zone (turbulent inner wall layer), cont.

1yUdf

dyτ

ν

−

+ ∝

11yUdf

dy yτκ

ν κ

−

+ + = =

or

where κ is a proportionality constant (known as Von Karman constant), hence

2 1UdU

dy yτ

ν κ +=1dU

dy yκ

+

+ +=or

• The integration gives the well=known “(semi) - logarithmic” velocity law:

(note: B=1/κ ln E)

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U+

log y+

U y+ +=1

ln( )U E yκ

+ +=

δν

viscousregion

1.0 yδ

1.0

0

UU

visc

ous

subl

ayer

buffe

r zo

ne

loga

rithm

ic r

egio

n

fully turbulent region

outer wall layer

inner wall layer

Velocity profile in a wall boundary layer, channel or pipe flow

nU y

U δ∞

=

Note: in the outer wall layer the effect of wall is minor; thus an external velocity and layer thickness are used for scaling, i.e.

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Turb. Wall boundary layers: Temperature field

PrT y+ += ⋅

( ) / /, ,p w w p pw w

T Tw

c T T c cy yf f

q

ρ τ ρ µ ρνρτ τ ρµ λ ν λ

−= =

ɺ

1 1ln (Pr) ln[ (Pr) ]

T T

T y B E yκ κ

+ + += + =

( ) ( , , , ,.., ,., .),w p wT wT T f y c q λτ ρ µ− = ɺ

( ,Pr)TT f y+ +=

• Following the same arguments one can derive universal distribution of temperature or any other scalar. e.g.

Molecular (conductive)

sublayerTurbulent wall zone (logarithmic layer)

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Thermal and hydrodynamic wall boundary layers:Effect of ( molecular) Prandtl number

Highly conductive fluids(liquid metals)

Common fluids(Air: Pr≈0.7-1.0Water: Pr≈7-12)

Low conducting fluids

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Thermal and hydrodynamic wall boundary layers:Effect of ( molecular) Prandtl number

Hot fluid /Cold wall

Cold fluid / Hot wall