FLOW NETS - Universiti Teknologi 2008-08-11آ  FLOW NETS For any two-dimensional irrotational flow...

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Transcript of FLOW NETS - Universiti Teknologi 2008-08-11آ  FLOW NETS For any two-dimensional irrotational flow...

  • IDEAL FLOW THEORY

    FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of lines may be drawn : (1) lines along which ψ is constant (2) lines along which φ is constant

    SECTION B 1

  • IDEAL FLOW THEORY

    stream line ψ perpendicular to the velocity potential φ These lines together form a grid of quadrilaterals having 90º corners. This grid is known as a flow net. It is provides a simple yet valuable indication of the flow pattern.

    SECTION B 2

  • IDEAL FLOW THEORY

    COMBINING FLOW PATTERNS

    If two or more flow patterns are combined, the resultant flow pattern is described by a stream function that at any point is the algebraic sum of the stream functions of the constituent flow at that point. By this principle complicated motions may be regarded as combinations of simpler ones.

    SECTION B 3

  • IDEAL FLOW THEORY

    21 ψψψψ +∆+=AP

    ψψψψ ∆++= 21AQ

    The resultant flow pattern may therefore be constructed graphically simply by joining the points for which the total stream function has the same value. This method was first described by W.J.M.Rankine (1820-1872)

    SECTION B 4

  • IDEAL FLOW THEORY

    Velocity components ;

    ( ) 212121 uuyyyyu +=∂ ∂

    + ∂ ∂

    =+ ∂ ∂

    = ∂ ∂

    = ψψψψψ

    21 vvx v +=

    ∂ ∂

    −= ψ

    Net velocity potential ;

    .......321 +++= φφφφnet

    SECTION B 5

  • IDEAL FLOW THEORY

    BASIC PATTERNS OF FLOW Uniform Flow ;

    velocity components ;

    α α

    sin cos ⋅= ⋅=

    qv qu

    stream function ;

    vxuy −=ψ velocity potential ;

    vyux +=φ

    SECTION B 6

  • IDEAL FLOW THEORY

    Source Flow ;

    A source is a point from which fluid issues uniformly in all directions. If for two-dimensional flow, the flow pattern consists of streamlines uniformly spaced and directed radially outward from one point in the reference plane, the flow is said to emerge from a line source.

    SECTION B 7

  • IDEAL FLOW THEORY

    The strength m of a source is the total volume rate of flow from it. The velocity q at radius r is given by;

    r mq π2 velocitylar toperpendicu area

    flow of rate volume ==

    velocity components ;

    0

    2

    = ∂ ∂

    −=′

    = ∂⋅

    ∂ =′

    r v

    r m

    r u

    ψ θ

    ψ

    stream function;

    π θψ

    2 m

    source =

    SECTION B 8

  • IDEAL FLOW THEORY

    velocity potential ;

    Crmsource += ln2π φ ( I ) at 00,0 =⇒== Crφ

    rmsource ln2π φ =

    ( II ) at A mCAr ln 2

    ,0 π

    φ −=⇒==

    ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛=

    A rm

    source ln2π φ

    SECTION B 9

  • IDEAL FLOW THEORY

    Sink ;

    A sink, the exact opposite of a source, is a point to which the fluid converges uniformly and from which fluid is continuously removed. The strength of a sink is considered negative, and the velocities, ψ , φ are therefore the same as those for a source but with the signs reversed.

    SECTION B 10

  • IDEAL FLOW THEORY

    stream function;

    π θψ

    2sink m

    −= velocity potential ;

    Crm +−= ln 2sink π

    φ ( I ) at 0,0 == rφ

    rm ln 2sink π

    φ −= ( II ) at Ar == ,0φ

    ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛−=

    A rm ln

    2sink π φ

    SECTION B 11

  • IDEAL FLOW THEORY

    Vortex ;

    2 types ; 1. Irrotational vortex 2. Forced vortex

    SECTION B 12

  • IDEAL FLOW THEORY

    Irrotational vortex ; Circulation ;

    δθδδ )(vortex rvvr ⋅′+′⋅=Γ

    rv π2vortex ⋅′=Γ vorticity ;

    0vortex = ′

    + ′

    = r v

    r v

    δ δζ

    stream function ;

    r r

    ln 2vortex π

    ψ Γ−= velocity potential ;

    θ π

    φ 2vortex Γ

    =

    SECTION B 13

  • IDEAL FLOW THEORY

    Forced vortex ;

    rv ⋅=′ ω vorticity ;

    ωςζ 20 =⇒≠

    r v

    r v

    δ δωζ ′

    + ′

    ==∴ 2

    SECTION B 14

  • IDEAL FLOW THEORY

    COMBINATION OF BASIC FLOW PATTERNS Linear and Source ;

    Stream function ;

    sourcelinearncombinatio ψψψ +=

    θ π

    θψ ⋅+⋅−= 2

    sin mrUncombinatio

    SECTION C 1

  • IDEAL FLOW THEORY

    stagnation point S is the point where the resultant velocity is zero.

    U mBOS π2

    == stream function at 0=θ ;

    0 2

    sin0 =⋅+⋅−== θπ θψψ

    mU It is called ‘stagnation line’. The body whose contour is formed by the combination of uniform rectilinear flow and a source is known as a half body, since it has a nose but no tail, or Rankine body.

    SECTION C 2

  • IDEAL FLOW THEORY

    Distance from origin to 0=ψ ;

    θπ θ sin2 ⋅

    = U mr

    Asymptote y ;

    ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛−=⋅=

    U m

    U mry

    2 and

    2 sinθ

    Velocity components ;

    θ π

    cos 2

    ⋅−=′ U r

    mu

    θsin⋅=′ Uv

    SECTION C 3

  • IDEAL FLOW THEORY

    If rectilinear flow comes from the other side ;

    2 m

    ncombinatio =ψ ( )

    θπ θπ

    sin2 ⋅ −

    = U

    mr

    θ π

    cos 2

    ⋅+=′ U r

    mu θsin⋅−=′ Uv

    SECTION C 4

  • IDEAL FLOW THEORY

    Source and Sink ;

    In this situation, the assumption again being made that the fluid extends to infinity in all directions.

    SECTION C 5

  • IDEAL FLOW THEORY

    Combination of stream function ; sinksource ψψψ +=ncombinatio

    ( )212 θθπψ −= m

    ncombinatio

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ +−

    = − 222 1 2tan

    2 yAx Aym

    ncombinatio π ψ Component velocity ;

    ( ) ( ) ⎥⎦ ⎤

    ⎢ ⎣

    ++ +

    − +−

    − = 22222 yAx

    Ax yAx

    Axmu π

    ( ) ( ) ⎥⎦ ⎤

    ⎢ ⎣

    ++ −

    +− = 22222 yAx

    y yAx

    ymv π

    velocity potential ;

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ ⋅=

    2

    1ln 2 r

    rm ncombinatio π

    φ

    SECTION C 6

  • IDEAL FLOW THEORY

    Source, Sink and Linear ;

    Combination of stream function ;

    linearncombinatio ψψψψ ++= sinksource

    Uy yAx

    Aym ncombinatio −⎥

    ⎤ ⎢ ⎣

    ⎡ ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ +−

    = − 222 1 2tan

    2π ψ

    SECTION C 7

  • IDEAL FLOW THEORY

    Component velocity ;

    ( ) ( ) U

    yAx Ax

    yAx Axmu −⎥

    ⎤ ⎢ ⎣

    ++ +

    − +−

    − = 22222π

    ( ) ( ) ⎥⎦ ⎤

    ⎢ ⎣

    ++ −

    +− −

    = 22222 yAx y

    yAx ymv

    π

    value of x ;

    1+= UA mAx

    π value of ymax ;

    ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛= −

    A y

    U my max1max tanπ

    SECTION C 8

  • IDEAL FLOW THEORY

    COMBINATION OF BASIC FLOW PATTERNS Doublet ;

    SECTION D 1

  • IDEAL FLOW THEORY

    Stream function ;

    ( ) θ π µθθ

    π ψ sin

    22 21 r m

    ncombinatio =−= velocity components ;

    θ π µ cos

    2 2r u =′

    θ π µ sin

    2 2r v =′

    22 r q

    π µ

    = velocity potential ;

    θ π µφ cos

    2 r −=

    SECTION D 2

  • IDEAL FLOW THEORY

    Doublet and Uniform ;

    Stream function ;

    θ π µψ sin

    2 ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛ −= Ur

    rncombinatio

    SECTION D 3

  • IDEAL FLOW THEORY

    velocity potential ;

    θ π µφ cos

    2 ⎟ ⎠ ⎞

    ⎜ ⎝ ⎛ +−= Ur

    rncombinatio

    0=ncombinatioψ , 0=θ , πθ =

    U r

    π µ

    2 =

    SECTION D 4

  • IDEAL FLOW THEORY

    U Ar

    π µ

    2 22 ==

    stream function ;

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ −−= 2

    2

    1sin r AUrncombinatio θψ

    velocity potential ;

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ −−= 2

    2

    1cos r AUrncombinatio θφ

    SECTION D 5

  • IDEAL FLOW THEORY

    velocity components ;

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ −−=′ 2

    2

    1cos r AUu θ

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ +=′ 2