A STUDY ON VISCOUS FLOW

(With A Special Focus On Boundary Layer And Its Effects)

COMPLETED BY RAJIBUL ALAM

M. Tech. (1st year) Roll No-14AE60R03

Viscosity is an aspect of friction. Whenever a fluid moves over a solid surface or over an another fluid , the mutual frictional resistance arising in the contact surfaces is known as viscosity and such a flow is known as viscous flow. The viscous shear stress is given by Ƭ=µ. where, Ƭ=shear stress µ=coefficient of viscosityFor air at standard room temperature , µ =1.7894* kg/ms FUNDAMENTAL EQUATION: Newton’s second law gives =m ------------- (1) where is the force in X direction, m is the mass and is the acceleration in X direction . Now consider a fluid element as shown in figure. Here shear stresses on three faces are shown.

Now considering forces on the fluid element in X direction only, we have = ( p- ( p + .dx ) ) .dy. dz + ( ( + .dx ) - ) .dy . dz + ( ( + .dy ) – ) ) .dx .dz + ( ( + .dz ) - ) ) .dx .dy On simplification which gives, = ( - + + ) .dx .dy. dz -------------------(2)

again mass m = ρ.dx.dy.dz where ρ is the density of the fluid

also = = + V. = + ( u + v + w )

Putting these values in equation (1) we have

- + + = ρ. + ρw.

This is the Navier-Stokes equation in X direction for a viscous flow . Similarly equations For other directions can also be found.

Types of viscous flows : Viscous flows can be differentiated into two types viz Laminar and Turbulent flow.

Laminar Flows : Laminar flows are those in which streamlines are smooth and regular and a fluid particle moves smoothly along a streamline.

Turbulent Flows : Turbulent flows are those in which streamlines break up and fluid particles move randomly in a zigzag fashion.

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Boundary Layer : When a solid body moves in viscous fluid or vice versa , the fluid layer adjacent to the solid body sticks to it . As we move perpendicular to the fluid motion from the solid body, the velocities of fluid layers keep on increasing till it reaches free stream fluid velocity . Thus a velocity gradient exists in the direction perpendicular to the fluid motion. This region where a velocity gradient exists in the direction perpendicular to the fluid motion is known as boundary layer.

Types Of Boundary Layer : a) Blasius Boundary Layer : It is the boundary layer attached to a flat plate held in oncoming unidirectional flow. b) Stokes Boundary Layer : It is a thin shear layer develops on an oscillatory body in a viscous liquid. In such a case when a fluid rotates, viscous forces are balanced by Coriolis effect. Coriolis effect is the deflection of a body when viewed in a rotating frame of reference.

BLASIUS B.L. STOKES B.L.

S

c) Laminar Boundary Layer: When a laminar flow takes place over a bounding surface , the associated boundary layer is known as Laminar Boundary Layer.

d) Turbulent Boundary Layer: When a turbulent flow takes place over a bounding surface the associated boundary layer is known as Turbulent Boundary Layer. Boundary Layer Properties: 1)Velocity Boundary Layer Thickness (∂): Boundary Layer thickness at a point on the solid surface is the height from that point to a point where velocity is 99% of the free stream velocity . Boundary layer thickness at different points may be different. 2)Thermal Boundary Layer Thickness ( ): Thermal boundary layer thickness at point on the solid surface is height from that point to a point where the fluid temperature is equal to the free stream flow temperature.

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3) Displacement Thickness ( ∂* ) : It is the height proportional to the missing mass flow rate due to the presence of boundary layer . It also gives the displacement through which a streamline drifts due to the presence of boundary layer.

Let u and ρ be the instantaneous velocity and density of a fluid inside a boundary layer . Let free stream values be . Then missing mass flow due to the presence of boundary layer - ------------(3)

Now if ∂* be the momentum thickness then, equation (3)= ..∂* This gives ∂* = 1- ).

Boundary layer equations: For a steady two dimensional flow continuity and momentum equations are given by, (ρ.u ) + ( ρ.v )=0 ------------------(4) u. + v. = - . + . ---------(5)

SOME RESULTS: (A) LAMINAR B.L. : For an incompressible flow over a flat plate ∂= ∂*=

For compressible flow

∂= ∂*=

(B) TURBULENT B.L. : For an incompressible flow over a flat plate

∂=

Aerodynamic Boundary Layer : It was first proposed by Ludwig Prandtl in a paper presented on August 12,1904 at the third International Congress Of Mathematics in Heidelberg,Germany. It simplifies equations of a flow around an airfoil by dividing the flow field into two areas viz flow adjacent to the airfoil i.e. inside the boundary layer where viscosity is dominant and outside the boundary layer where the flow can be assumed to be inviscid. Viscous flow solution: This differentiation of flow leads to a simplified solution to viscous flow problems . INVISCID FLOW

VISCOUS FLOW

IN

(a) First for the flow outside the boundary layer inviscid solution is carried out and ,and are found at the outer extreme of the boundary layer.

(b) Using the above values the boundary layer equations (4) and (5) are solved and then momentum thickness ∂* is found. Once momentum thickness is found effective body shape i.e. body + boundary layer is determined. Now within this region viscous calculations are carried out.

Effects of types of boundary layer on shear stress : Velocity profile of a laminar boundary layer is completely different from that of a turbulent boundary layer. Incase of a laminar boundary layer, velocity gradually reduces from free stream value at the outer layer to zero at the wall while incase of a turbulent boundary layer velocity is almost uniform near to the solid surface and suddenly becomes zero as evident from the following graph.

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Pressure distribution in a boundary layer: Pressure through a boundary layer in a direction perpendicular to the surface is constant.

Thus in figure though pressure increases in the direction of the flow, but it remains constant in a given section, viz 1-1, 2-2 etc. This is an important phenomenon. This is why pressure calculated from inviscid calculation gives correct value for real life surface pressure.

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Explanation:

For full presentation clickhttp://rajibulalam.blogspot.in/2016/03/a-study-on-viscous-flow-with-special.html u. / + v. / = - /𝝏𝒖 𝝏𝒙 𝝏𝒖 𝝏𝒚 𝟏 ρ . ( / ) + µ . ( ^ )/( .y^ )𝝏𝒑 𝝏𝒙 𝝏 𝟐 𝒖 𝝏 𝟐 Now at y=0, v=0 and =0 . hence, 𝟏/ρ . ( / ) = µ . ( ^ )/( .y^ )𝝏𝒑 𝝏𝒙 𝝏 𝟐 𝒖 𝝏 𝟐 however boundary layer suction is provided then equation becomes

v. / + /𝝏𝒖 𝝏𝒚 𝟏 ρ . ( / ) = µ . ( ^ )/( .y^ )𝝏𝒑 𝝏𝒙 𝝏 𝟐 𝒖 𝝏 𝟐 𝟏/ρ . ( / ) = µ . ( ^ )/( .y^ )𝝏𝒑 𝝏𝒙 𝝏 𝟐 𝒖 𝝏 𝟐 Now if / =0 ,then ( ^ )/( .y^ ) =0 at the wall i.e. is at a maximum there 𝝏𝒑 𝝏𝒙 𝝏 𝟐 𝒖 𝝏 𝟐and falls away steadily.

If however > 0 i.e. if there is an adverse pressure gradient ( ^ )/( .y^ ) > 0 i.e. is at a 𝝏 𝟐 𝒖 𝝏 𝟐minimum there. Hence /( ) first increases and then decreases with y. At this point flow is 𝝏𝒖 𝝏𝒚at the verge of separation.

If pressure gradient further increases velocity profile gets distorted and ( /( ) )y=0 𝝏𝒖 𝝏𝒚becomes zero first and flow is at the verge of separation now. On further increase of pressure gradient( /( ) )y=0 becomes negative and flow reversal happens.𝝏𝒖 𝝏𝒚

WAKE REGION PRESSURE REDUCES SHARPLY

Wake due to separation

How does lift reduce due to separation : It is now clear that due to separation , pressure downstream an airfoil on the upper surface reduces sharply which increases drag drastically.But at the same time any decrease in pressure on the upper surface should increase the lift.But why does the lift decrease due to separation?

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IMPORTANCE OF INVISCID FLOW: Though the concept of perfect inviscid fluid is absurd, yet in many cases viscosity associated is negligible. In such cases inviscid flow calculation gives almost accurate results.

Moreover in case of viscous flow over an aerofoil the flow can be divided into two regime as discussed earlier, viz viscous and inviscid which makes the life easier.

Most importantly , some recent research by NASA is hinting that even as complex problem as flow separation till date to be thought of completely viscous dominated phenomenon may be in reality be an inviscid dominated flow which requires only a rotational flow. For example some inviscid flow field numerical solutions for flow over a circular cylinder when vorticity is introduced by means of a curved shock wave , accurately predicting the separated flow on the rear side of the cylinder.

Boundary layer and shock wave interaction :

Boundary layer does not mixwith a shock wave, rather whena viscous supersonic flow takesplace over a flat plate shockwave impinges on the boundary layer. Due to strong adverse pressure gradient across a shock wave the boundary layer gets separated . Since high pressure behind the shock feeds upstream through the subsonic portion of the boundary layer , hence separation takes place ahead of the theoretical inviscid flow impingement point of the shock wave . In turn separated boundary layer deflects the oncoming flow into itself , thus creating a second oblique shock wave known as leading edge shock wave. Separated boundary layer subsequently reattaches the surface at some downstream position. Here the flow is again turned into itself creating a third shock wave known as reattachment shock wave. In between the leading edge shock and reattachment shock supersonic flow is turned away from itself through some expansion fans. The scale and severity of the interaction depends on the type of boundary layer. A laminar boundary layer more readily separates than a turbulent boundary layer .

Boundary layer-shock wave interaction in a nozzle : We know for pressure ratio higher than some critical value normal shock wave stands right inside the nozzle as shown in the adjacent figure.

However the study of Craig A. Hunter presented in a paper called “Experimental Investigation Of Separated Nozzle Flows “ shows that normal shock wave interacts with boundary layer inside the nozzle to cause local separation of the flow and again reattachment of the flow. This causes the formation of Lamda type shock wave as shown in the adjacent figure. LAMDA TYPESHOCK WAVE

REFERENCES : A) Fundamentals Of Aerodynamics --by John D Anderson Jr B) Principle Of Flight --by John D Anderson Jr C) Edinburgh University Publication On Boundary Layer D) Internet

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