Gradually Varied Flow I+II - Lunds tekniska hأ¶ Gradually Varied Flow. ... Computation of Gradually

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  • Gradually Varied Flow I+II

    Hydromechanics VVR090

  • Gradually Varied Flow

    Depth of flow varies with longitudinal distance.

    Occurs upstream and downstream control sections.

    Governing equation:

    21 −

    = − o fS Sdy

    dx Fr

    (previously Sf = 0 was studied)

  • Derivation of Governing Equation

    Total energy:

    2

    2 uH z y g

    = + +

    Differentiating with respect to distance:

    ( )2 / 2 = + +

    d u gdH dz dy dx dx dx dx

  • = −

    = −

    f

    o

    dH S dx

    dz S dx

    For a given flow rate:

    ( )2 2 2 2 3 3

    / 2d u g Q dA dy Q T dy dyFr dx gA dy dx gA dx dx

    = − = − = −

    (slope of energy grade line)

    (bottom slope)

    21 −

    = − o fS Sdy

    dx Fr Resulting equation:

  • Definition of Water Surface Slope

    Water surface slope dy/dx is defined with respect to the channel bottom.

    Hydrostatic pressure distribution is assumed (streamlines should be reasonably straight and parallel).

  • • The head loss for a specific reach is equal to the head loss in the reach for a uniform flow having the same R and u. Manning equation yields.

    • The slope of the channel is small

    • No air entrainment

    • Fixed velocity distribution

    • Resistance coefficient constant in the reach under consideration

    2 2

    4/3f n uS R

    =

    Assumptions made when solving the gradually varied flow equation:

  • Classification of Gradually Varied Flow Profiles

    The following conditions prevail:

    If y < yN , then Sf > So If y > yN , then Sf < So

    If Fr > 1, then y < yc If Fr < 1, then y > yc

    If Sf = So , then y = yN

  • Water surface profiles may be classified with respect to:

    • the channel slope

    • the relationship between y, yN , and yc .

    Profile categories:

    • M (mild) 0 < So < Sc • S (steep) So > Sc > 0

    • C (critical) So = Sc • A (adverse) So < 0

  • Gradually Varied Flow Profile Classification I

  • Gradually Varied Flow Profile Classification II

  • Mild Slope (M-Profiles)

    Profile types:

    1: y > yN > yc => So > Sf and Fr < 1

    => dy/dx > 0

    2: yN > y > yc => So < Sf and Fr < 1 => dy/dx < 0

    3: yN > yC > y => So < Sf and Fr > 1 => dy/dx > 0

    0 < So < Sc

  • Steep Slope (S-Profiles)

    Profile types:

    1: y > yc > yN => So > Sf and Fr < 1

    => dy/dx > 0

    2: yc > y > yN => So > Sf and Fr > 1 => dy/dx < 0

    3: yc > yN > y => So < Sf and Fr > 1 => dy/dx > 0

    0 < Sc < So

  • Final Form of Water Surface Profile

    1. y Æ •, Sf Æ 0, Fr Æ 0, and dy/dx Æ So

    2. y Æ yN , Sf Æ So , and dy/dx Æ 0

    3. y Æ yc , Fr Æ 1, and dy/dx Æ •

    21 −

    = − o fS Sdy

    dx Fr

    Asymptotic conditions:

  • Transition from Subcritical to Supercritical Flow

  • Transition from Supercritical to Subcritical Flow

  • Example: Flow into a Channel from a Reservoir

  • Flow Controls

    • determine the depth in channel either upstream or downstream such points.

    • usually feature a change from subcritical to supercritical flow

    • occur at physical barriers, for example, sluice gates, dams, weirs, drop structures, or changes in channel slope

    Locations in the channel where the relationship between the water depth and flow rate is known (or controllable).

    Controls:

  • Strategy for Analysis of Open Channel Flow

    1. Start at control points

    2. Proceed upstream or downstream depending on whether subcritical or supercritical flow occurs, respectively

    Typical approach in the analysis:

  • Computation of Gradually Varied Flow

    21 −

    = − o fS Sdy

    dx Fr Governing equation:

    Solutions must begin at a control section and proceed in the direction in which the control operates.

    Gradually varied flow may approach uniform flow asymptotically, but from a practical point of view a reasonable definition of convergence is applied.

  • Uniform Channel

    Prismatic channel with constant slope and resistance coefficient.

    Apply energy equation over a small distance Dx:

    2

    2 o f d uy S S dx g

    ⎛ ⎞ + = −⎜ ⎟

    ⎝ ⎠

    Express the equation in difference form:

    ( ) 2

    2 o f uy S S x g

    ⎛ ⎞ Δ + = − Δ⎜ ⎟ ⎝ ⎠

  • Over the short distance Dx assume that Manning’s equation is suitable to describe the frictional losses (Sf ):

    2 2

    4/3f n uS R

    =

    The equation to be solved may be written:

    ( ) ( )

    2

    2 2 4 /3

    / 2

    /o mean

    y u g x

    S n u R

    Δ + Δ =

  • Dxi

    Reach i

    x

    yi yi+1

    ( ) ( ) ( )

    2 2

    1 2 2 4/3

    1/ 2

    / 2 / 2

    / i i

    i o i

    y u g y u g x

    S n u R +

    +

    + − + Δ =

    All quantities known at i. Assume yi+1 and compute Dxi (ui+1 given by the continuity equation).

    ui ui+1

  • Example 6.1

    A trapezoidal channel with b = 6.1 m, n = 0.025, z = 2, and So = 0.001 carries a discharge of 28 m3/s. If this channel terminates in a free overfall, determine the gradually varied flow profile by the step method.

    b = 6.1 m 2

    1yN

  • Solution:

    Compute normal water depth.

    ( )

    ( )

    2/3

    2

    2

    1

    2 1

    2 1

    o

    N N

    N

    N N

    N

    Q AR S n

    A b zy y

    P b y z b zy y

    R b y z

    =

    = +

    = + +

    + =

    + +

    yN = 1.91 m

  • Compute critical water depth:

    ( )

    1 /

    2

    c c c

    c c c

    c

    u QFr gD A gA T

    A b zy y T b zy

    = = =

    = +

    = +

    yc = 1.14 m

    yN > y > yc Mild slope (yN > yc )

    M2 profile

  • Table for step calculation:

    y A P R u u2/2g Sf Sfav Dx S

    (Dx)

    1.14 9.55 11.20 0.85 2.93 0.438 0.0067

    0.0058 3 3

    1.24 10.64 11.64 0.91 2.63 0.353 0.0049

    0.0044 9.3 12.3

    1.32 11.54 12.00 0.96 2.43 0.300 0.0039

    and so on ( ) ( )2 2 1

    , 1/ 2

    / 2 / 2 i i

    i o f i

    y u g y u g x

    S S +

    +

    + − + Δ =

    ( ), 1/ 2 , 1 ,12f i f i f iS S S+ += + 2 2

    4/3f n uS R

    =

  • Other Solution Methods

    Problem with the step method is that the water depths is obtained at arbitrary locations (i.e., the water depth is not calculated at fixed x-locations).

    By direct integration of the governing equation this problem can be circumvented.

    Different approaches for direct integration:

    • semi-analytic

    • trial-and-error

    • finite difference

  • Semi-Analytic Approach

    Find solution in terms of closed-form functions (integrals).

    Employ suitable approximations to these functions or some look-up tables.

    Approach OK for channels with constant properties.

    (for more information, see French)

  • Trial-and-Error Approach

    Well-suited for computations in non-prismatic channels.

    Channel properties (e.g., resistance coefficient and shape) are a function of longitudinal distance.

    Depth is obtained at specific x-locations.

    Apply energy equation between two stations located Dx apart (z is the elevation of the water surface):

    2

    2 2 1 2

    1 2

    2

    2 2

    f e

    f e

    uz S x h g

    u uz z S x h g g

    ⎛ ⎞ Δ + = − Δ −⎜ ⎟ ⎝ ⎠

    + = + + Δ +

    he : eddy losses

  • Equation is solved by trial-and-error (from 2 to 1):

    1. Assume y1 Æ u1 (continuity equation)

    2. Compute Sf (and he , if needed)

    3. Compute y1 from governing equation. If this value agrees with the assumed y1 , the solution has been found. Otherwise continue calculations.

    Estimate of frictional losses:

    ( )1 212f f fS S S= +

  • Example 6.4

    A trapezoidal channel with b = 20 ft, n = 0.025, z = 2, and So = 0.001 carries a discharge of 1000 ft3/s. If this channel terminates in a free overfall and there are no eddy losses, determine the gradually varied flow profile by the trial-and-error step method.

    b = 20 ft 2

    1yN

  • Solution Table

    Stn. z y A u u2/2g H1 R Sf Sfav Dx hf H2

    0 103.74 3.74 103 9.71 1.46 105.20 2.81 0.00670 105.20

    116 104.62 4.50 130 7.69 0.92 105.54 3.24 0.00347 0.00509 116 0.590 105.79

    105.02 4.90 146 6.85 0.73 105.75 3.48 0.00251 0.00461 116 0.535 105.73

    355 105.56 5.20 158 6.33 0.62