Force On Immersed Bodies

Force on Immersed Bodies 1. Introduction: In engineering fields there are various problems which involve the fluid around the submersed bodies. In such problems either a fluid may be flowing around submerged stationary body or body may be flowing through a large mass of stationary fluid. Examples: •

Motion of very small objects such as sand particles in air or water.

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Large bodies such as airplane, submarines, automobiles, ships etc moving through air or water

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Structure such as buildings and bridges etc which are submerged in air or water.

2. Force Exerted by a Flowing Fluid on a Body: Whenever there is relative motion between a real fluid and a body, the fluid exerts a force on a body and the body exerts equal and opposite force on the fluid. A body fully immersed in a real fluid may be subjected to two kinds of forces called drag force and lift force. Drag force: The component of force in the direction of flow on a submerged body is called drag force (FD). Lift force: The component of force in the perpendicular to the flow is called the lift force (FL). In the symmetrical body moving through an ideal fluid (no viscosity) at a uniform velocity, the pressure distribution around a body is symmetrical and hence the resultant force acting on the body is zero. However real fluids such as air, water, posses viscosity and if it is moved through these fluid at a uniform velocity, the body experiences a resistance to motion. For the symmetrical body such as sphere and cylinder facing the flow is symmetrical, there is no lift force. For the production of lift force there must be asymmetry of flow, but drag force exists always. It is possible to create drag without lift but impossible to create lift without drag. The fluid viscosity affects the flow around the body causing the force on the body accordingly. At low Reynolds' Number the fluid is deformed in very wide zone 1

Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

around the body causing pressure force & friction force. As Reynolds' Number increases, viscous effects are confined to the boundary layer causes predominant the friction force on the boundary.

3. Expressions for Drag & Lift: Pressure and friction forces on in elementary surface of an immersed body

Consider a body held stationary in a stream of real fluid moving at a uniform velocity U. Let Θ = inclination of the tangent to the small element dA with the direction of flow. Then the force acting on dA of the surface of the body can be considered to have two components τdA (called shear force) and pdA (called pressure force) acting along the directions tangential and normal to the surface respectively. The tangential components are called shear force and the normal components are called pressure forces. The summation of component of the forces acting over the entire surface of the body in the direction of fluid flow is drag force, FD and perpendicular to fluid flow is lift force, FL FD = ∫ap dA sin θ + ∫aτodA cosθ FL = ∫aτodA sin θ - ∫ap dA cosθ Here, ∫ap dA sin θ is called the pressure drag and ∫aτodA cosθ called the friction drag or skin drag or shear drag.

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Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

The contribution of shear stresses to the lift may be neglected since shear stresses are small as compared.

3. B Calculation of the drag and the lift forces: For the body moving through a fluid density ρ at a uniform velocity U, the mathematical expression for the calculation of the drag and the lift forces are given by FD = CD A and FL = CLA Where, CD = coefficient of drag CL = coefficient of lift A = characteristics area = area projected on a plane perpendicular to the relative motion of the fluid, in the case of calculating FD. = area projected on a plane perpendicular to the direction of lift force, in the case of calculating FL. Note: •

The lift force may exist even in ideal fluid by the presence of circulation.

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Real fluid also requires vortices or circulations around the body for producing lift.

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Drag force is caused by inertia, viscosity, wave action of free surface, gravity.

4. Pressure Drag and Friction Drag: The contribution of pressure drag and friction drag to the total drag depends on the flowing parameters. •

Characteristics of fluid

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shape of body Orientation of the body immersed in the fluid.

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Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

When a thin plate is placed parallel to the direction of flow the pressure drag will be zero & the total drag is entirely due to shear stress, thus equal to friction drag.

When the same plate is held with its axis normal to flow direction the friction drag will be zero. In this case the total drag is due to the pressure force only.

5. Friction drag of Boundary layer - Incompressible flow: 6.Boundary Layer: When real fluid pass a solid boundary, a layer of fluid comes in contact with the boundary surface due to viscosity. This layer of fluid cannot slip away from the boundary surface and it has the same velocity as that of the boundary. At the boundary surface there is no relative motion between the fluid & the boundary. This condition is known as no sleep condition. If the boundary is stationary the fluid velocity at the boundary surface will be zero. Thus at the boundary surface the layer of fluid undergoes retardation. This retarded layer of the fluid further causes the retardation of the adjacent layers of the fluid. The velocity of the flowing fluid increases gradually from zero at the boundary surface to the velocity of the main stream. This region is known as boundary layer. The large variation of velocity in a relatively small distance exists large velocity gradient (dv/dy) normal to the boundary surface which results the corresponding share stress (is of considerable magnitude. Away from the boundary layer retardation due to viscosity is negligible and the velocity there will be equal to that of the main stream. The resistance due to 4

Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

viscosity is confined only in the boundary layer. The fluid outside the boundary layer may be considered as ideal.

Near the leading edge of a plate, the boundary layer is fully laminar. For a boundary layer the velocity distribution is parabolic. Thickness of boundary layer increases with the distance from the leading edge as more and more fluid is slowed down by the viscous boundary, become unstable and breaks into turbulent boundary layer over a transition region.

7. Expression for friction drag at boundary layer: 1.Control volume for flow over one side of a flat plate

Thickness of boundary layer = δ (at the distance x from the edge) is the distance from the boundary in which velocity reaches 99% velocity of the free stream. Considering the control volume ABCD at distance x along the plate and along control surface AB undisturbed velocity U exists and assume there is absence of the pressure force around the periphery of the control volume. According to the impulse momentum principle -Fx

= - Drag 5

Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

= [rate of momentum in x direction leaving through BC + rate of momentum in x direction leaving through AB] - [rate of momentum in x direction entering through DA ]......(i) Since discharge through BC is less than DA, there is flow out of control volume across control surface AB. i.e. QAB = QDA – QBC Let b be the width of the plate then flow rate and momentum across control surfaces are:

This assumption is valid when 1.There is no pressure gradient along the surface 2. Its boundary layer doesn't change from laminar to turbulent within the region considered.

8. Boundary layer separation and pressure Drag: The forces acting on the boundary layer are, 1.Inertia force 2. Viscous force 3.Pressure force When pressure decreases in the direction of flow, the flow is accelerated and then the pressure gradient (dp/dx) exists. If this is less than 0 both inertia and pressure force tends to reduce the effect of boundary layer in the direction of flow which reduces the losses in the boundary layer.

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Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

When pressure increases in the direction of flow (dp/dx >0), the pressure force acts opposite to the direction of the flow and further increases the retarding effect of the viscous force. That increases in the thickness of the boundary layer in the direction of flow. If these forces act over a long stretch, the boundary layer gets separated from the surface and moves into the main stream. This phenomenon is called separation. The point at which the boundary layer is separate from the surface is called the point of separation.

9. Effect of pressure gradient on boundary layer development: So far, we have assumed zero pressure gradient in the flow direction across the flat surfaces considered. In the curved surface, there is the presence of pressure gradient dp/dx which effectively means a du/dx term, i.e., the flow stream velocity is seen to vary as show in the figure below. if pressure is decreased in downstream direction, boundary layer thickness tends to reduce. if pressure is increased in downstream direction, boundary layer thickens rapidly increase.

9 B. Separation of boundary layer:

Consider a fluid flows round the surface (the area of flow decreases).It is accelerated over the right hand section until at point B the velocity just outside the boundary is maximum and the pressure is minimum. From A and B the pressure gradient is negative. As long as dp/dx <0, the entire BL moves forward.

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Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]

Beyond B, the area of flow increase and hence velocity of flow decreases, due to decrease of velocity, the pressure increases in the direction of flow and hence the pressure gradient (dp/dx) is positive.

The value of velocity gradient (du/dx) at the boundary is zero at point C, this is known as separation point, where the BL starts separating from the surface because further retardation of flow near the surface is physically impossible. Growth and separation of boundary layer owing to increasing pressure gradient Large turbulent eddies are formed downstream of the point of separation the disturbed region in which eddies are formed is called Turbulent Wake. The flow separation depends upon curvature of the surface Reynolds' Number of flow roughness of the surface

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Presented by: Dr. Rajendra Shrestha [HOD, Mechanical Engineering, IOE, Pulchowk]