hypersonic ae3021_f06_10

download hypersonic ae3021_f06_10

of 29

Transcript of hypersonic ae3021_f06_10

  • 8/6/2019 hypersonic ae3021_f06_10

    1/29

    Hypersonic flow: introduction

    Van Dyke: Hypersonic flow is flow past a body at high Mach number, wherenonlinearity is an essential feature of the flow.

    Also understood, for thin bodies, that if is the thickness-to-chord ratio of the body,M is of order 1.

    Special Features

    Thin shock layer: shock is very close to the body. The thin region between the shock andthe body is called the Shock Layer.

    Entropy Layer: Shock curvature implies that shock strength is different

    for different streamlines stagnation pressure and velocity gradients -rotational flow

  • 8/6/2019 hypersonic ae3021_f06_10

    2/29

    http://www.onera.fr/conferences/ramjet-scramjet-pde/images/hypersonic-funnel.gif

    The Hypersonic Tunnel For Airbreathing Propulsion

  • 8/6/2019 hypersonic ae3021_f06_10

    3/29

    Velocity-Altitude Map For Re-Entry

    Velocity

    Altitude

    Typical re-entry case:Very little deceleration untilVehicle reaches denser air

    (Deliberately so - to avoidlarge fluctuations in aerodynamicloads and landing point )

  • 8/6/2019 hypersonic ae3021_f06_10

    4/29

    Atmosphere

    Troposphere: 0 < z < 10km

    Stratosphere: 10 < z < 50km

    Mesosphere: 50 < z < 80km

    Thermosphere: z > 80km

    Ionosphere 65 < 365 km Contains ions and free electrons

    60

  • 8/6/2019 hypersonic ae3021_f06_10

    5/29

    A Simple Model for Variation of density with altitude

    gdz dp !

    M T R

    p

    =

    Neglect dissociation and ionization Molecular weight is constant Assume isothermal (T = constant) poor assumption

    dz T R

    M g

    p

    dp

    !

    !"

    #$%

    &' z

    T R M g

    e

    log0

  • 8/6/2019 hypersonic ae3021_f06_10

    6/29

    Non-lifting body moving at velocity V, which is inclined at angle to the x-axis:

    !DCosdt

    x d m "

    2

    2

    mg DSindt

    z d m !"2

    2

    mg S C U dt z d

    m D !"sin21 2

    2

    2

    !"

    #$%

    &S C

    m

    Dis the Ballistic Parameter.

    Assuming that the drag force is >> weight and that is constant because gravitational force istoo weak to change the flight path much

    !"#

    $%&''!

    "

    #$%

    &RT gMz

    mS C

    U U

    Log De

    e expsin21 0

    (

    )

    U

    D

  • 8/6/2019 hypersonic ae3021_f06_10

    7/29

    www.galleryoffluidmechanics.com/shocks/s_wt.htm

    High Angle of Attack Hypersonic Aerodynamics

  • 8/6/2019 hypersonic ae3021_f06_10

    8/29

    http://www.scientificcage.com/images/photos/hpersonic_flow.jpgy

  • 8/6/2019 hypersonic ae3021_f06_10

    9/29

    Croccos Theorem:

    !rr

    "uhsT 0

    Viscous Layer:

    Implies vorticity in the shock layer.

    Thick boundary layer, merges with shock wave to produce a merged shock-viscous layer.Coupled analysis needed.

    High Temperature Effects:

    Very large range of properties (temperature, density, pressure) in the flowfield, so thatspecific heats and mean molecular weight may not be constant.

    Low Density Flow:

    Most hypersonic flight (except of hypervelocity projectiles) occurs at very high altitudes

    Knudsen No. =

    L

    != ratio of Mean Free Path to characteristic length

    Above 120 km, continuum assumption is poor. Below 60 km, mean free path is less than 1mm.

  • 8/6/2019 hypersonic ae3021_f06_10

    10/29

    http://www.aerospace-technology.com/projects/x43/images/X-43HYPERX_7.jpg

  • 8/6/2019 hypersonic ae3021_f06_10

    11/29

    Summary of Theoretical Approaches

    Newtonian Flow: Flow hits surface layer, and abruptly turns parallel to surface.

    Normal force decomposed into lift and drag.Modified Newtonian Flow: Account for stagnation pressure drop across shock.

    Local Surface Inclination Method : Cp at a point is calculated from static pressure behind an obliqueshock caused by local surface slope at freestream Mach number.

    Tangent Coneapproach: similar to local surface slope arguments.

    Mach number independence: Shock/expansion relations and Cp become independent of Machnumber at very high Mach number.

    Blast wave theory: Energy of Disturbance caused by hypersonic vehicle is like a detonation wave.Hypersonic similarity: Allows developing equivalent shock tube experiments for hypersonicaerodynamics.

  • 8/6/2019 hypersonic ae3021_f06_10

    12/29

    Local Surface Inclination Methods Approximate methods over arbitrary configurations, in particular,

    where Cp is a function of local surface slope.Newtonian Aerodynamics

    Newton (1687) concept was that particles travel along straight lines withoutInteraction with other particles, let pellets from a shotgun. On striking a surface,they would lose all momentum perpendicular to the surface, but retain all tangential momentum i.e., slide off the surface.

    In 3D flows we replace

    ASinU !" #22Net rate of change of momentum

    !22 SinCp =

    !SinU " with nU

    rr

    !

    2

    2

    2!

    !

    =

    U

    nU Cp

    r

    Shadow region: 0Cp

    Shadow region is where 0! nU r

    r

  • 8/6/2019 hypersonic ae3021_f06_10

    13/29

    Remarks on Newtonian Theory:

    Poor in low speed flow. Predicts .2

    !l C

    (1) Works well as Mach number gets large and specific heat ratio tends towards 1.0Why? Because shock is close to surface, and velocity across the shock is very large most of thenormal momentum is lost.

    (2) Tends to overpredict c p and c d (C D) see figure 3.11

    (3) Works better in 3-D than in 2-D(4) In 3-D, works best for blunt bodies; not good for wedges, cones, wingsetc.

  • 8/6/2019 hypersonic ae3021_f06_10

    14/29

    Was proposed by Lester Lees in 1955, as a way of improving Newtoniantheory, and bringing in Mach Number and dependence on

    . He proposed replacing 2 with

    !M pC

    max pC

    !2

    maxsin p p C C =

    Here is the coefficient behind a Normal shock wave,at the stagnation point. That is,

    max pC pC

    2

    02

    max

    21

    !

    !=

    U

    p pC

    p#

    Modified Newtonian

  • 8/6/2019 hypersonic ae3021_f06_10

    15/29

    From Rankine-Hugoniot relations,

    ( )( ) !

    !"

    #

    $$%

    &

    +

    +

    !!"

    #

    $$%

    &

    '

    += (

    (

    (

    ( 1

    21

    124

    1 21

    2

    2202

    )

    )

    )

    ) ))

    M

    M

    M p p

    (3.17)

    Then

    2

    02

    2

    1

    !

    !

    "

    =

    M

    p p

    c p #

  • 8/6/2019 hypersonic ae3021_f06_10

    16/29

    In the limit as ,!M We get

    ( )

    ( )1

    4

    4

    1

    1

    12

    +!

    "

    #$$

    %

    & +=

    '

    '

    ((

    (

    ((

    ((

    pc

    As ,4.1 839.1max ! pc

    As ,1 2max=

    pc Proposed by Newton

    Exercise: Compute c p values for configurations shown on Figures 3.8,3.6, 3.11 and 3.12 using Newtonian and Modified Newtonian theories.Biconvex Airfoil.

    y/c = 0.05 -0.2 (x/c) 2

  • 8/6/2019 hypersonic ae3021_f06_10

    17/29

    Where does freestream Mach number appear in the above?Only in the dependence of downstream pressure, density, temperature.

    As freestream Mach number becomes large,( )( )1

    1

    1

    2

    !

    +"

    #

    #

    $

    $

    !"#$

    %&

    !"#

    $%&

    +

    =

    ''

    '

    '

    '2

    222

    22

    2 1sin1

    2

    M M

    U p

    p p

    U p

    ()

    ((

    *

    !"

    " 2sin1

    2

    +=

    Why nondimensionalize by2

    !

    U

    Because ( )22 ~ !U O p " And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Machnumber does not appear anywhere.

    Mach Number Independence

  • 8/6/2019 hypersonic ae3021_f06_10

    18/29

    The blast wave theory argues that the sudden addition of energy to thefluid by the body is equivalent to a high explosive of energy E being

    exploded at time t=0.

    A shock wave associated with the explosion spreads away from the originwith time

    In 2-D problem: the shock wave is a plane wave:

    !

    =

    U

    x t

    Shock wave moves outward with tBlast wave origin

  • 8/6/2019 hypersonic ae3021_f06_10

    19/29

    Hypersonic Shock & Expansion Relations

    Why?

    1. Simpler than exact expressions - for analysis2. Key parameter is seen to be M where is the flow turning angle, for M>>1 and

  • 8/6/2019 hypersonic ae3021_f06_10

    20/29

    !"

    #$%

    &'

    +

    + 1sin1

    21 221

    1

    2(

    )

    )M

    p p

    !"

    #$%

    &'

    +

    + 11

    21 221 )

    *

    *M

    ( ) ( )2

    222

    1

    2 1

    4

    1

    4

    11

    K K K

    p p

    +"#

    $%& ++

    ++

    '

    Defining pressure coefficient

    21

    1

    2

    2

    1

    M

    p p

    C p!

    "#$

    %&' (

    )

    !!"

    #

    $$%

    &+

    ()

    *+, +

    ++

    =

    '()

    *+, -

    . 2

    2

    2

    1

    2

    2

    1

    4

    1

    4

    12

    2

    1

    K K

    p p

    C p //0

  • 8/6/2019 hypersonic ae3021_f06_10

    21/29

    Next

    ( )( ) 21

    221

    1

    2

    1

    1sin1

    M

    M

    v

    u

    +

    !!"

    #

    In the hypersonic limit,

    1

    sin21

    2

    1

    2

    +

    !

    #

    $v

    u

    Also

    ( )( ) 21

    221

    1

    2

    1

    1sin2

    M

    Cot M

    v

    v

    +

    !

    = "

    #

    ( )12sin

    1

    2

    +!

    "

    #v

    v

  • 8/6/2019 hypersonic ae3021_f06_10

    22/29

    Density Jump Across Shock

    ( )( ) 2sin1

    sin122

    1

    221

    1

    2

    +

    +=

    "

    "

    $

    $

    M

    M

    In the hypersonic limit, for large M 1 >>1, finite

    ( )( )1

    1

    1

    2

    !

    +"

    #

    #

    $

    $

    Then

    ( )( )222

    112

    12

    12

    1

    sin12

    +

    !=

    "#

    $$ M p pT T

  • 8/6/2019 hypersonic ae3021_f06_10

    23/29

    21

    1

    2

    2

    1

    M

    p p

    C p !

    "#

    $%&

    '(

    )

    1

    4 2

    +=

    !

    "SinC p 11 > >M

    Hypersonic Shock Relations in the Limit of Large but FiniteMach number and small turning angle

    We define a similarity parameter !1M K = which can be used to collapse avariety of data

  • 8/6/2019 hypersonic ae3021_f06_10

    24/29

    ( ) 2cos1sin

    cot2tan22

    1

    221

    +!="

    ""

    M

    M

    For large but finite M, small and

    becomes

    ( )!!"

    #$$%

    &+

    ++

    +' 22

    1

    2 1

    16

    1

    4

    1

    (

    )

    (

    *

    M

    Works for finite values of M1 = K

  • 8/6/2019 hypersonic ae3021_f06_10

    25/29

    Hypersonic Expansion Wave Relations

    From Prandtl-Meyer theory, 12 !#

    ( ) ( )1tan11

    1tan

    1

    1 2121!"

    #

    $%&

    '()

    *+,

    -!

    !+

    !+

    = ! M M .

    .

    .

    ./

    For 11 > >M 212

    1 1 M M !

    Also ( ) !"#

    $%&' '

    x x

    1tan

    2tan 11

    (

    From Taylor series

    ..5

    1

    3

    111tan

    531

    !#$

    %&'

    x x x x

  • 8/6/2019 hypersonic ae3021_f06_10

    26/29

    2

    1

    1

    11

    21

    1 !

    "

    "

    "

    "# $%

    &

    '()

    *$+

    $$+

    +M M

    ( ) 212

    11

    2!

    "

    "# $$$+= M

    Then

    ( ) !"#

    $%&

    '' 21

    12

    11

    1

    2

    M M )

    ( )( )

    1

    22

    21

    1

    2

    11

    11 !

    ""#

    $

    %%&

    '

    +

    +=

    (

    (

    (

    (

    M

    M p p

    1

    2

    2

    1 !"#$%

    &'( )

    )

    M M

    1

    2

    1

    2

    11

    2

    2

    11

    2

    11 !

    "#$

    %&' !!"#

    $%&' !!

    (

    (

    (

    (

    ()

    (K M

    p p

    !!!

    "

    #

    $$$

    %

    &'!"

    #$%& ''

    ()*

    +,- '

    .' 1

    2

    11

    2

    2

    11

    2

    22

    1

    2

    2/

    /

    /

    /0K

    K K

    p p

    C p ),(2 !"K f

    C p#

    Note that

  • 8/6/2019 hypersonic ae3021_f06_10

    27/29

    Consider flow over a blunt body:

    Where does freestream Mach number appear in the above?Only in the dependence of downstream pressure, density, temperature.

    As freestream Mach number becomes large,( )( )1

    1

    1

    2

    !

    +"

    #

    #

    $

    $

    !"#$

    %&!

    "#$

    %&

    +

    =

    ''

    '

    '

    '2

    222

    22

    2 1sin1

    2

    M M

    U p

    p p

    U p

    ()

    ((

    *

    !"

    " 2sin1

    2

    +=

    Why nondimensionalize by2

    !U

    Because ( )22 ~ !U O p " And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Machnumber does not appear anywhere.

    Mach Number Independence

  • 8/6/2019 hypersonic ae3021_f06_10

    28/29

    This Mach number independence is also observed in experiments. Sphere drag coefficient,for example.

  • 8/6/2019 hypersonic ae3021_f06_10

    29/29

    Hypersonic Aerodynamics Roadmap

    Supersonic Aero

    Local SurfaceInclinationMethods

    Blast WaveTheory

    Newtonian Aerodynamics Newton

    Buseman

    HypersonicSmall Disturbance:Mach Number Independence

    Full shock-expansion methodWith real gas effects

    Stagnation Point: CFD

    Conical Flow /Waveriders

    Non-Equilibrium Gas Dynamics