ME 6139: High Speed Aerodynamicstoufiquehasan.buet.ac.bd/ME 6139 (October 2019)-L-02.pdf · ME 6139: High Speed Aerodynamics 6 Source: T. A. Ward, Aerospace Propulsion Systems, John

ME 6139: High Speed Aerodynamics 1

ME 6139: High Speed Aerodynamics

Dr. A.B.M. Toufique HasanProfessor

Department of Mechanical Engineering, BUET

Lecture-0226 February 2020

[email protected]

M.Sc. Eng., ME, BUET (Oct. 2019)

High speed flows & Compressibility

M.Sc. Eng., ME, BUET (Oct. 2019) ME 6139: High Speed Aerodynamics2

Compressibility of a fluid is basically a measure of the change of density that willbe produced in the fluid by a specific change in pressure.In a fluid flow there are usually changes in pressure associated, for example, withchange in velocity in the flow. These pressure changes will in general induce densitychanges, which will have an influence on the flow.

If these density changes are important, the temperature changes are becomingalso important.

The study of flows in which the changes in density and temperature are importantis basically what is known as Aerothermodynamics, compressible fluid flow orgas dynamics.

It usually only being in gas flows that compressibility effects are important.

High speed flows deals with the fluid velocity which is atleast comparable to thespeed of sound. In this massive velocity, the change of fluid density becomesvery significant and must be accounted for the prediction of other fluid dynamicand thermodynamic property changes. Compressibility plays a great role.

Mach Number

M.Sc. Eng., ME, BUET (Oct. 2019) ME 6139: High Speed Aerodynamics3

aVM

whereV is the flow velocity and a is the local speed of sound in the fluid.

In fluid mechanics, the effect of compressibility in the flow field can be assessedby a number called the “Mach number”. This dimensionless number is defined as

Why speed of sound ? ? ?

This is the speed at which “signal” (disturbance) can travel through the medium. Incase when an object moves through a fluid, it generates disturbances (infinitesimalpressure waves, which are sound waves) that emanate from the object in all director.When the speed of object becomes comparable or higher than the speed of sound,then the propagation and interaction of disturbance (signal) become complicated anddifferent compared to low speed cases.

Flow Classification


In aerodynamics, the following flow classes are classified roughly depending onthe Mach number-

M < 0.3 : Incompressible flow (density effects are negligible)

0.3< M < 0.8 : Subsonic flow, where density effects are important butno appearance of shock waves

0.8< M < 1.2 : Transonic flow, where shock waves first appear, dividing thesubsonic and supersonic flows.

1.2 < M < 3.0 : Supersonic flow, where shock waves are present but thereare no subsonic regions.

M > 3.0 : Hypersonic flow, where shock waves and other flow changesare especially strong.(Surface Chemistry, Plasma dynamics)


Applications of High speed flows


The most obvious applications of compressible flow theory are in the design of highspeed aircraft. These includes:

• Commercial civil aircraft• Military fighters• Ramjet vehicle• Scramjet vehicle• Rockets etc.

However, the knowledge of compressible fluid flow theory is required in the design andoperation of many devices commonly encountered in engineering practice. Amongthese applications are:

• Gas turbines: The flow in the blades and nozzles is compressible.• Steam turbines: here the flow in the nozzles and blades must be treated ascompressible.• Reciprocating engines: the flow of the gases through the valves and in theintake and the exhaust systems must be treated as compressible.• Combustion chambers: the study of combustion, in many cases requires aknowledge of compressible fluid flow.


Flight Envelope


Source: T. A. Ward, Aerospace Propulsion Systems, John Wiley & Sons (Asia), Pte Ltd., Singapore (2010)

Engines can operate only over a certain range of altitudes and velocities (Mach numbers)which correspond to differing atmospheric pressure, temperature and densities (allchanges with altitude). This range is known as the engine’s flight envelope.


Atmosphere


Reynolds number and Mach number significantly vary with altitude at the same flow velocity.


Shock Waves


Shock waves are obvious in internal or external aerothermodynamics.

The extremely thin region in which the transition from the initial supersonic velocity(M>1), relatively low-pressure state to the state that involves a relatively low velocity(M<1) and high pressure is termed as a normal shock wave.

The thickness is usually only a few mean free paths.

Supersonic initial flow (M>1) is mandatory for the generation of a shock wave.


Physical Examples


shadowgraph of supersonic flowaround space crew modules: Mach2.2 flow around an Apollo-like capsuleat 25° angle-of-attack

the essential ingredients of these flowsincluding: presence of multiple shockwaves, separated zones, and wakes, andlarge scale structures.*

shocks

*Source: T. B. Gatski & J P Bonnet, Compressibility, Turbulence and High Speed Flow, Elsevier, The Netherlands (2013)



Physical Examples contd…

Flow, M = 2.3


Shock wave/boundary layer interaction (SWBLI)

EF



M>1

Shock waves in Compression ramp flow




Flow




Interaction of shock waves in compressor cascade


ME 6139: High Speed Aerodynamics14

Conservation equationsIn integral forms


Continuity Equation


Physical PrincipleMass can be neither created nor destroyed.Apply this principle to the model of a fixed control volumein a flow as shown in figure. The volume is V, and the areaof the closed surface is S.Consider point B on the control surface and an elementalarea around B is ds. Let n be the unit vector normal to thesurface at B. Define dS = nds. Also let, V and ρ be thelocal velocity and density at B.Letting denote the mass flow rate through ds, andreferring to the fig.

m

SV ddsVdsVm n )cos(

The net mass flow into the control volume through the entire control surface S is the sum of the elemental mass flows, namely-

)(idS SV

where the minus (-ve) sign denotes inflow (opposite to the direction shown in fig.)

Volume, V


(Vn is the velocity normal to the surface ds)

Continuity Equation contd…


Now, consider an infinitesimal volume, dV inside the control volume. The mass ofthis infinitesimal volume is ρdV. Hence, the total mass inside the control volumeis the sum of these elemental masses, namely-

The time rate of change of this mass inside the control volume is therefore-

V

Vd

)(VV

iidt

Finally, the physical principle that mass is conserved states that the net mass flowinto the control volume must equal the rate of increase of mass inside the controlvolume. So, the mathematical presentation in the form of integrals is-

V

Vdt

dS

SV

This equation is called the continuity equation; it is in the integral formulation ofthe conservation of mass principle and applicable to all flows, compressible orincompressible, viscous and inviscid.


Momentum Equation


Physical PrincipleThe time rate of change of momentum of a body equals the net force exertedon it. This statement in vector form-

First consider, the forces (F) on the control volume.Forces (F) can be of two types-

FV mdtd

1. Body forces acting on the fluid inside V. These couldbe gravitational force, electromagnetic forces that may beexerted on the fluid inside V. Let f represent the bodyforce per unit mass of fluid (N/kg). Considering anelemental volume dV, the elemental body force is (ρdV)f.Hence, summing over the complete control volume-

VforcebodyTotalV

d f


Momentum Equation contd…


2. Surface forces acting on the boundary of the controlvolume. There are two sources of surface forces in afluid element namely- pressure and shear forces.Dealing with the inviscid flows (μ = 0), the only surfaceforce is therefore due to pressure. Consider elementalarea dS, the surface force acting on this area is –pdS,where the minus sign signifies that pressure actsinward. Hence, summing over the complete controlsurface

S

pdSpressuretodueforcesurfaceTotal

At any given time instant, the total force F acting on the control volume is-

)(VV

idpdρS SfF

body force surface force




The rate of change of momentum has two components

1. Let A1 represents the net rate of flow of momentumacross the surface S. The elemental mass flow acrossds is given by . With this elemental mass flow isassociated a momentum flow

SV d VSV d

The net rate of flow of momentum, summed over thecomplete surface S is,

S

d VSV1A

2. Consider, an unsteady flow, where the flow properties at any given point inthe flow field are functions of time (such as flow over oscillation body, nozzleflows with valve on and off). If the control volume is drawn in such an unsteadyflow, then the momentum of the fluid inside the control volume would befluctuating with time simply due to the time variations in ρ and V. In this case,A1 does not represent the whole contribution to the rate of change of momentum.There is, in addition, a time rate of change of momentum due to unsteady,transient effects in the flow field inside V.


Vmdtd



Let A2 represents fluctuation in momentum. Alsoconsider an elemental mass of fluid, ρdV. The mass hasthe momentum (ρdV)V. Summing over the completecontrol volume V

V

VV inside momentum Total dρV

Hence, the change in momentum in V due to unsteadyfluctuation in the local flow properties is

VV

2 V)(V dtρdρ

tA VV

The sum A1 and A2 represents the total instantaneous time rate of change ofmomentum of the fluid as it flows through the control volume-

)(V)(

V21 iid

tρdAAm

dtd

S

VVSVV




Finally, relate the physical principle in mathematical form gives-

FV )(mdtd

This is the momentum equation in integral formulation applied to inviscid fluidflows.

viscousVV

VV)( FSfVVSV

SS

dpddtρd

If friction were to be included, it would appear an additional surface force,namely, shear and normal viscous stresses integrated over the control surface. IfFviscous represents this surface integral, then the modified form of momentumequation is-

SS

dpddtρd SfVVSV

VV

VV)(


One-Dimensional Steady Flow


In many practical situations, it is adequate toassume the flow is steady and one-dimensional (1D).This assumption can be applicable withacceptable accuracy to flows with a variablearea provided that the rate of change of areaand the curvature of the system are smallenough for one component of the velocityvector to remain dominant over the other twocomponents. For example, although the flowthrough a nozzle of the type shown in figure isnot strictly one-dimensional.

However, since the v (y-component of V) remains very much less than u (x-component of V) then the flow can be calculated with sufficient accuracy for mostpurposes by ignoring v and assuming the flow is 1-D.

Further, assumption of steady flow usually gives a good description of the meanflow values of the flow variables.



Continuity equationConsider the flow through a one-dimensionalregion as represented by the shaded area in thefigure. A control volume is taken and there are fourcontrol surfaces (left, right, top and bottom).Also assume that the flow is steady; then theintegral form of the continuity equation becomes as-

Evaluating the surface integral gives- (left and right surfaces give –ρ1v1A1 andρ2v2A2, respectively, whereas top and bottom surfaces contribute nothing to thesurface integral because V and dS are perpendicular to each other)

0

VV

S

S

d

dt

d

SV

SV

= 0

constant

0

222111

222111

VAAVAVAVAV

Continuity equation for steady 1-D flow

One-Dimensional Steady Flow contd…


Documents

ME 6139: High Speed Aerodynamicstoufiquehasan.buet.ac.bd/ME 6139 (October 2019)-L-02.pdf · ME 6139: High Speed Aerodynamics 6 Source: T. A. Ward, Aerospace Propulsion Systems, John