SOLUTION OF BOUNDARY LAYER

SOLUTION OF BOUNDARY LAYER

EQUATIONS

EQUATIONS

Prabal Talukdar

Prabal Talukdar

Associate Professor

Department of Mechanical Engineering

Department of Mechanical Engineering

IIT Delhi

Boundary layer Approximation

Applying Newton‘s 2nd law in the y-direction, we get y-momentum equation P 2 _∂ ∂ ⎞ ⎛ ∂ ∂ x P y u y u v x u u ₂ 2 ∂ ∂ − ∂ ∂ μ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ X momentum: y P x v y v v x v u 2 2 ∂ ∂ − ∂ ∂ μ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ 0 P ₌ ∂

)

x

(

P

P

,

Thus

=

dx

dP

x

P

,

Hence

=

∂

∂

0 y ∂

For a flate plate, since u = U_∞ = constant and v = 0 outside the boundary layer,

X-t ti i

0

x

P

=

∂

∂

momentum equation gives

Therefore, for flow over a flat plate, the 

pressure remains constant over the entire 

p

plate (both inside and outside the boundary 

Paul Richard Heinrich

The continuity and momentum equations 

were first solved in 1908 by the German 

i H Bl i t d t f L

Paul Richard Heinrich

Blasius (1883 – 1970) was a German Fluid Dynamic

Engineer. He was one of the

engineer H. Blasius, a student of L. 

Prandtl.

This was done by transforming the two 

first students of Prandtl.

Born 9 August 1883

B li G _{partial differential}y  _{equations into}g  _a single 

ordinary differential equation by 

introducing a new independent variable, 

called the similarity variable

Berlin, Germany

Died 24 April 1970 (aged 86) Hamburg, West Germany

called the similarity variable. 

The finding of such a variable, assuming it 

exists, is more of an art than science, and 

Fields Fluid mechanics and  mechanical engineering Alma mater University of Göttingen

it requires to have a good insight of the 

problem.

Doctoral 

The shape of the velocity profile remains the same along the plate.

Blasius reasoned that the nondimensional velocity profile u/u

_∞

should

remain unchanged when plotted against the nondimensional distance

y/

δ

where

δ

is the thickness of the local velocity boundary layer at a

y/

δ

,

where

δ

is the thickness of the local velocity boundary layer at a

given x.

That is, although both

,

g

δ

and u at a given y vary with x, the velocity u at

g

y

y

,

y

a fixed y/

δ

remains constant

Blasius was also aware from the work

of Stokes that

δ

is proportional to

ν

x

∞

Scale Analysis

Similarity Variable

The significant variable is y/δ, and we

assume that the velocity may be expressed as a function of this variable. We then haveu ct o o t s va ab e. We t e ave

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ν

=

η

∞

u

x

/

y

define

We

δ

≈

η

y

/

,

makes

This

Here, η is called the similarity variable, and g(η) is the function we seek as a solution

Variable Transformation

A stream function was defined such that:

to get rid of continuity equation

u ∂Ψ ∂Ψ to get rid of continuity equation y u ∂ = x v ∂ − = x u y ν = η ∞

udy

d

Ψ

=

( )

∞ ∞ ν η = Ψ u x u f x u dy d ν = η ∞

∫

η η = η) g( )d ( f where

( )

( )

_{∞ ∞} ∞ ∞ ν η = Ψ ⇒ ν Ψ = η f u x u u x u f ⎟⎟ ⎞ ⎜⎜ ⎛ − η ν = ν − η ∂ ν − = Ψ ∂ − = η = ν η ν = ∂ η ∂ η ∂ Ψ ∂ = ∂ Ψ ∂ = ∞ ∞ ∞ ∞ ∞ ∞ ∞ f df u 1 f u df x u v d df u x u d df u x u y y u P.Talukdar/Mech-IITD ⎟⎟ ⎠ ⎜⎜ ⎝ η η ∂ η ∂x u ∞ u _∞ d x 2 u _∞ x f 2 x d f v

Differentiating the previous equation with respect to x and y 3 2 2 2 2 2 f d u u f d u u , d f d x 2 u x u ∂ ∂ η η − = ∂ ∂ ∞ ∞ ∞

Substituting these relations into the momentum equation and 3 2 2 , _{y x d} d x u y = ν _{η ∂} = ν _η ∂ ∞ ∞ ∞

Substituting these relations into the momentum equation and simplifying, we obtain 0 d f d f d f d 2 2 2 3 3 = η + η

This is a third-order nonlinear differential equation.

This way the system of two PDEs is converted to one ODE

η η

Blasius Solution

0 d f d f d f d 2 ₂ 2 3 3 = η + η

The value of η corresponding to u/u_∞ = 0.992 is 5.0

x

u

5

x

u

y

ν

δ

=

⇒

ν

=

η

∞ ∞

0

5

0

5

_Significance of u_∞, ν, x x

Re

x

0

.

5

x

u

0

.

5

=

ν

=

δ

∞

The shear stress at the wall can be

d

i d f

determined from:

2 2 w d f d x u u y u _∞ ∞ η ν μ = ∂ ∂ μ = τ 0 0 y x d y = η = ν η ∂ 2 u 332 . 0 u 332 0 ρμ ∞ ρ ∞

Local skin friction coefficient becomes

x w Re x u 332 . 0 _∞ ρμ ∞ = ρ ∞ = τ

Local skin friction coefficient becomes

2 1 x 2 w 2 w x , f 0.664Re 2 u 2 V C − ∞ = ρ τ = ρ τ =

Note that unlike the boundary layer

thickness, wall shear stress and the skin

∞

ρ ρ

P.Talukdar/Mech-IITD

friction coefficient decrease along the plate

as x

-1/2

.

Energy Equation

gy q

Introduce a non-dimensional temperature

s s T T T ) y , x ( T ) y , x ( − − = θ ∞

Substitution gives an energy equation of the form:

s T T_∞ 2 2 y y v x u ∂ θ ∂ α = ∂ θ ∂ + ∂ θ ∂

Temperature profiles for flow over an isothermal flat plate are similar like the velocity profiles.

Thus, we expect a similarity solution for temperature to exist.

Further, the thickness of the thermal boundary layer is proportional to

ν

x

/

u

_∞ Using the chain rule and substituting the u and v expressions into the energy equation gives

Using the chain rule and substituting the u and v expressions into the

energy equation gives

2 2 2 y d d y d d f d df x u 2 1 x d d d df u _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ η ∂ θ α = ∂ η ∂ η θ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − η η ν + ∂ η ∂ η θ η ∞ ∞

Compare

df/d

η

is replaced by

θ

2 _y d y d d x 2 x d dη η ∂ ⎜_⎝ η ⎟_⎠ η ∂ _η ⎜_{⎝ ∂} ⎟_⎠ 0 f d f f d 2 2 3 = + 0 d d f Pr d 2 2 2 = θ + θ

For Pr = 1

2 dη3 + f dη2 = 0 d dη2 η 1 d df and 0 d df 0 = η = η _{η= η=∞}

( )

0 = 0 θ

( )

∞ = 1 θ and

Thus we conclude that the velocity and thermal boundary layers coincide, and 

the nondimensional velocity and temperature profiles (u/u and θ) are identical

0 η=∞

= η

the nondimensional velocity and temperature profiles (u/u_∞ and θ) are identical 

for steady, incompressible, laminar flow of a fluid with constant properties and 

Pr = 1 over an isothermal flat plate

P.Talukdar/Mech-IITD

The value of the temperature gradient at the surface (Pr =1) ?? 332 . 0 d f d d d 2 2 = η = η θ

This eq. is solved for numerous values of Prandtl numbers. For Pr > 0.6, the nondimensional temperature gradient at

1/3 0 d d f Pr d d 2 2 2 = η θ + θ Pr > 0.6

the surface is found to be proportional to Pr1/3

d dη 2 η 3 1 Pr 332 . 0 d d = η θ 0 dη _η= s s T T T ) y , x ( T ) y , x ( − − = θ ∞ x u y y ν = δ = η ∞ d T ∂θ θ ∂ ∂

The temperature gradient at the surface is

(

)

(

)

y d d T T y T T y T s s _∂∂θ = − _ηθ ∂_∂η − = ∂ ∂ = = η ∞ = ∞ =

The local convection coefficient can be expressed as: u k Pr 332 0 y T k q h s ∂ y 0 = 1 3 ∂ − = = = ∞ _{0 332 P}

(

_{T T}

)

u y T 3 1 0 y = ∂ ∂ ∞ =

And the local Nusselt number becomes

(

T T

)

(

T T

)

0.332 Pr k x h s s x = ₋ = ₋ = _ν ∞ ∞ 0.332 Pr

(

T Ts

)

_x 3 1 ν − ∞ ∞ 2 1 x 3 1 x x h_kx 0.332 Pr Re Nu = = Pr > 0.6

Solving the thermal boundary layer equation numerically for the temperature profile for different Prandtl numbers, and using the definition of the thermal boundary layer, it is determined thaty y ,

3 1 Pr

≅ δ

Non-dimensionalization

Non dimensionalization

0 v u ₌ ∂ ∂ + ∂ ∂ y x ∂ ∂ p u u v u u 2 _∂ − ∂ μ = ⎟⎟ ⎞ ⎜⎜ ⎛ ∂ ₊ ∂ ρ x y y v x u ₂ ∂ − ∂ μ = ⎟⎟ ⎠ ⎜⎜ ⎝ ∂ + ∂ ρ ⎟ ⎞ ⎜ ⎛ _∂ + ∂ ⎟⎟ ⎞ ⎜⎜ ⎛ ∂ + ∂ 2T 2T k T T ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ ∂ + ∂ = ⎟⎟ ⎠ ⎜⎜ ⎝ ∂ + ∂ ρ _{p 2 2} y x k y v x u c s s * 2 * * * * * T T T T T and V p p , V v v , V u u , L y y , L x x − − = ρ = = = = = ∞

v u* ∂ * ∂ 0 y v x u * * _∂ = ∂ + ∂ ∂ * * 2 * * * *

∂

u

∂

u

1

∂

u

dp

Continuity: * 2 * L * * * *

dx

dp

y

u

Re

1

y

u

v

x

u

u

−

∂

∂

=

∂

∂

+

∂

∂

Momentum: 2 * * 2 L * * * * * * y T Pr Re 1 y T v x T u ∂ ∂ = ∂ ∂ + ∂ ∂ Energy:

With the boundary conditions:

( )

0

y

1

u

( )

x

0

0

u

(

x

)

1

v

( )

x

0

0

u

*

( )

*

=

*

( )

*

=

*

(

*

∞

)

=

*

( )

*

=

( )

0

,

y

1

,

T

( )

x

,

0

0

,

T

(

x

,

)

1

T

0

0

,

x

v

,

1

,

x

u

,

0

0

,

x

u

,

1

y

,

0

u

* * * * * *

=

=

∞

=

=

=

∞

=

=

Geometrically Similar

Geometrically Similar

Functional forms of Friction and

C

i

C ffi i

Convection Coefficients

* * 2 * * * * u _v u 1 u dp u ∂ + ∂ ∂ Momentum:

For a given geometry the solution for u*

* 2 * L * * _dx y Re y v x u − ∂ = ∂ + ∂ Momentum:

(

)

For a given geometry, the solution for u*

can be expressed as

Th

h h

h

f

b

(

* * L

)

1 * _{f x ,y ,Re} u =

Then the shear stress at the surface becomes

(

* L

)

2 0 y * * 0 y s _LV f x , Re y u L V y u * μ = ∂ ∂ μ = ∂ ∂ μ = τ = =0 _{y 0} y y ₌

* 2 * * * * ∂T ∂T 1 ∂ T

The solution for T* can be expressed as

2 * L * * * * y T Pr Re 1 y T v x T u ∂ ∂ = ∂ ∂ + ∂ ∂ Energy:

(

x ,y ,Re ,Pr

)

g T* = ₁ * * _L

Using the definition of T*, the convection heat transfer coefficient becomes

k ) T T ( k y T k∂ ∂ − N lt b 0 y * * 0 y * * s s s 0 y * * _L T y k y T ) T T ( L ) T T ( k T T y T k h = = ∞ ∞ ∞ = ∂ ∂ = ∂ ∂ − − − = − ∂ ∂ = hL Nusselt number: Pr) , Re , x ( g y T k hL Nu ₂ * _L 0 y * * x = = ∂ ∂ *₌ =

Note that the Nusselt number is equivalent to the

dimensionless temperature gradient at the

surface and thus it is properly referred to as the

surface, and thus it is properly referred to as the

dimensionless heat transfer coefficient

Average friction coefficient Average friction coefficient

Reynold Analogy

Reynold Analogy

When Pr =1 (approximately the case for gases) and ∂P*/ ∂x* = 0 (e g For flat

2 * * 2 L * * * * * * y u Re 1 y u v x u u ∂ ∂ = ∂ ∂ + ∂ ∂

for gases) and ∂P / ∂x = 0 (e.g. For flat

plate) y 2 * * 2 L * * * * * * y T Re 1 y T v x T u ∂ ∂ = ∂ ∂ + ∂ ∂ * T hL Nu = = ∂ 0 y * x * y k Nu = ∂ = = x * * 0 s Nu L V y u L V y u * μ = ∂ ∂ μ = ∂ ∂ μ = τ _{* *} T u ∂ ∂ 0 y 0 y y y _{∂ *} ∂ = = x L 2 x 2 s x , f Nu Re 2 2 V Nu L V 2 V C = ρ μ = ρ τ = 0 y * 0 y * * * y y ₌ = ∂ ₌ ∂ x L x , f Nu 2 Re C = _(Pr=1) L Re 2 V 2 V ρ ρ x x , f St 2 C = (Pr=1) 2 Pr Re Nu V c h St L p = ρ =

Clinton-Colburn Analogy

Also called modified Reynolds analogy

2 1 R 664 0 C − N 0 332 P 1 3 R 1 2 Colburn j-factor 2 1 x x , f 0.664Re C = Nu _x = 0.332 Pr 1 3 Re _x1 2 C Taking the ratio between C_f,x and Nu_x

Valid for 3 1 x x x , f Nu Pr 2 Re C = − 23 H p x x , f j Pr V c h 2 C ≡ ρ =

Although this relation is developed using relations for laminar flow over a flat plate (for which

∂P*/ ∂ x* = 0), experimental studies show that it is l li bl i t l f t b l t fl

0.6<Pr<60

also applicable approximately for turbulent flow over a surface, even in the presence of pressure