prev

next

of 70

View

118Download

6

Embed Size (px)

Aerodynamics Lecture NotesSudhir Joshi Click to edit Master subtitle style

2/5/13

Viscous Flow

In reality every flow in the world is a viscous flow. Viscosity is the phenomena of friction that acts on all objects and fluids. In viscous flow, friction of surfaces, heat transfer and energy transfer between molecules and mass transfer (diffusion) takes place.2/5/13

Examples of Viscous Flow

2/5/13

Examples of Viscous Flow

2/5/13

Examples of Viscous Flow

2/5/13

Examples of Viscous Flow

2/5/13

Examples of Viscous Flow

2/5/13

Examples of Viscous Flow

2/5/13

Friction (Viscosity) Creates Shear Stress

2/5/13

The influence of friction creates V=0 at the body surface and this is called no-slip condition.

No Slip Condition

2/5/13

Adverse Pressure Gradients produces If the flow on the surfacean increasing pressure distribution in the flow direction (P3 > P2 > P1) then such a region is called an adverse pressure gradient.

This can create REVERSED FLOW in the field

2/5/13

Reversed Flow

When the fluid element is retarded by friction and increasing pressure, it will slow down and eventually stop as it travels downstream. Then, as pressure mounts, it will reverse direction and move back upstream (Reversed Flow)

2/5/13

Flow Separation

Reversed flow will cause the flow to separate from the surface and create a large wake of recirculating flow downstream of the surface. The point of separation occurs at dV/dN = 0

2/5/13

Flow Separation

2/5/13

Flow Separation & Airfoil Stall Separation of flow increases the dragand results in substantial loss of lift. Thus analysis of viscous flow and flow separation is important in order to prevent airfoil stall.

2/5/13

Flow Separation & Airfoil Stall

2/5/13

Effects of Viscosity on Shearbe seen from the Stress Hence, as it canslides above, viscosity can cause reversed flow as well as flow separation which can effect the stability of the airfoil.

2/5/13

Types of Drag Due to Viscous types of drag on Flow There are two mainthe airfoil surface due to Viscous flow effects. These are Df which is Skin friction drag Dp which is the pressure drag of separation These numbers will effect whether we want turbulent or laminar flow for better control of the airfoil surface 2/5/13

Heat Transfer in Viscous Flow a certain The moving fluid hasamount of energy. As it flows on the surface, the flow velocity is decreased due to friction.

This decrease in velocity is translated as a decrease in kinetic energy. This loss of kinetic energy transforms into internal energy of the fluid. (Conservation of Energy) Hence, the temperature of the fluid 2/5/13 will rise.

Aerodynamic Heating in Viscous Flow As a result, warmer fluid will heat thecooler surface of the body. This is called aerodynamic heating. This becomes more severe since as flow velocity increases aerodynamic heating will also increase.

2/5/13

Laminar and Turbulent Flow

It is very important to understand the effects of viscous flow on laminar and turbulent flow If the flow is smooth, it is called laminar flow. If the flow is irregular, random and jagged, it is turbulent flow.

2/5/13

Turbulent Viscous Flow

In turbulent flow, the frictional effects are more severe. Also, in turbulent flow, shear stress and aerodynamic heating are greater.

2/5/13

Laminar and Turbulent Flow

2/5/13

Turbulent Viscous Flow

However for turbulent flow, there is an advantage as the flow separation is less likely to happen. Even if flow separation occurs, the separation region will be much slower. The pressure drag Dp is smaller in turbulent flow.

2/5/13

Laminar and Turbulent Flow There are certain conditions whichcause laminar flow to become transitional and turn into turbulent flow. These reasons include: a) Increased Surface Roughness

b) Increased turbulence in the free stream c) Adverse pressure gradients d) Heating of the fluid by the surface will cause turbulent flow2/5/13

Transition to Turbulent Flow

2/5/13

When is Turbulent Flow There are Desirable? some cases whereturbulent flow may be desirable instead of laminar flow. For example for slender bodies, laminar flow is desirable, while for blunt bodies turbulent flow may be desirable. Usually Re>500,000 = Turbulent2/5/13

Flow

When is Turbulent Flow Desirable?

2/5/13

Effects of Heat on Viscous Flow

The moving fluid has a certain amount of kinetic energy as it flows on the surface. However, the flow velocity is decreased due to friction. This decrease in the viscous flow is the loss of kinetic energy of the flow which transforms into internal energy of the fluid. This causes the temperature to rise 2/5/13 the fluid and this is called viscous in

Effects of Heat on Turbulent Flow Turbulent flows can be more effectedby aerodynamic heating. If the wall of the surface is being heated that can cause the laminar flow to transform into turbulent flow. Consequently, if the wall surface of the airfoil is cooler that may cause the turbulent flow to subside and become laminar flow. Thus for blunt bodies, turbulent flow 2/5/13

When is Turbulent Flow Desirable?

2/5/13

Viscous Flow and Turbulent Flow flows. All viscous flows are rotationalThus no velocity potential function exists for viscous flow. In viscous flows, the Prandtl number is important. Prandtl number is the property of the gas and it changes as a function of temperature.

2/5/13

Boundary Conditions for Viscous exists at the Flow 1) No slip conditionairfoil surface. Hence fluid velocity is zero at the surface where y=0 >>>> u=v=w=0

2) The fluid will have the same temperature as the wall at the surface y=0 >>> T=Tw 3) If Tw is not constant, this means that either the fluid is heated by the wall or the wall is being heated by 2/5/13 the fluid. Then the boundary

How to Analyze a Viscous Flow? 1) Take the boundary conditionsgiven at the previous slide into consideration 2) Analyze the direction of the flow. Are there any elements that can be omitted? 3) Use the Equations given in the next section for calculating velocity and temperature gradients 4) Use Shear Stress and Fourier Formula for calculating shear stress 2/5/13 and heat flux caused by the fluid on

Fluid Element in a Viscous Flow

2/5/13

Forces Acting on the Fluid Element All of the forces acting on the Fluid

element can be summarized by Newtons Second Equation:Now lets rewrite the left hand side of the equation for all the forces acting on the x direction:

The right hand side of the equation is:2/5/13

Navier Stokes Equations

Navier- Stokes equations are the most famous equations that depict aerodynamic viscous flow. In essence, Navier Stokes Equations are the momentum-continuity-energy equations derived in the slide from Newtons Second Law.

Navier Stokes equations can be applied to any kind of Fluid flow. They depict all of the momentum forces acting on the various variables 2/5/13 of the fluid.

Navier Stokes Momentum Equations

2/5/13

Navier Stokes Momentum Equations

2/5/13

Energy Equations

2/5/13

2/5/13

Graphic Representation of a Navier Stokes Solution

2/5/13

Solutions to the Viscous Flow In order to solve a viscous flowvariables, you first need to write the Continuity Equation, Momentum Equations as well as the Energy Equation relevant to the flow at hand.

You simplify some terms thinking about the terms that may be cancelled such as delta t=0 for steady flow or 1 d flow with v=w=0 etc. 2/5/13

Modeling of the Navier Stokes Equations

2/5/13

Shear Stress in a Fluid Element

2/5/13

Shear Stress

2/5/13

Shear Stress

2/5/13

Basic Solutions to Viscous solutions to Flow There are two basicViscous Flow Equations. These two solutions are solved for analytically by using the Navier Stokes equations. These unsolvable equations are solved by making some terms come to zero due to cancellation of forces. The two main basic solutions to Viscous flow are:

2/5/13

Couette Flow

The Couette flow can be best described as the flow between two plates, where the top plate is moving and the bottom plate is stationary.

2/5/13

Couette Flow

The driving force for this flow is the motion of the upper plate dragging the flow along with it through the mechanism of friction. The upper plate is is exerting a shear stress acting toward the fluid at y=D that is causing the fluid to move toward the right.

2/5/13

Couette Flow

The only changes in the flow take place in the y direction. Nothing changes in the x and z direction. The flow velocity is zero at the bottom where the stationary plate is. The flow velocity gradually increases as d is increased and reaches maximum at the top where the moving plate is.

2/5/13

Couette Flow

The kinetic energy of the flow is dissipated partially through friction and turned to internal energy in the fluid. This is called viscous dissipation. The heat flux is expressed by the Fourier Law.

2/5/13

If the stationary plate and the

In this flow, all flow properties only change with y. Thus, all partial derivatives in the Navier Stokes equations with respect to x and z are zero. The flow is steady, so all time derivatives are zero.

Couette Flow

2/5/13

2/5/13

Navier Stokes Momen