Numerical Hydraulics Block 3 Open channel flow Markus Holzner
Contents of the course Block 1 The equations Block Computation of pressure surges Block 3 Open channel flow (flow in rivers) Block 4 Numerical solution of open channel flow Block 5 Transport of solutes in rivers Block 6 Heat transport in rivers
Saint Venant equations in 1D b(h) l b A(h) continuity momentum equation z h 3
Saint Venant equations in 1D continuity (for section without inflow) Q A + = 0 x t Momentum equation from integration of Navier-Stokes/Reynolds equations over the channel cross-section: v v h P τ l ρ + ρα v = ρg t x x A 0 b 4
Saint Venant equations in 1D Friction force: Only the shear stress at the channel walls survives integration. Per unit volume it is f friction force τ0lbδx τ0 = = = volume AΔx R For the wall shear we can insert hy λ τ ρ λ λ 8 ( ) 0 = v = (Re, k/ R hy ) 5
Saint Venant equations in 1D The friction can be expressed as energy loss per flow distance: ΔE ΔE/ V τ 0 = freibung Δx = V Δx R Using friction slope and channel slope hy I E Δ τ = = = = Δ E/ V 0 λ 1 v z IS xρg ρgrhy 4 Rhy g x Alternative: Strickler/Manning equation 6
Saint Venant equations in 1D we finally obtain v v h + v = g( I ) S IE g t x x ( va( h)) A( h) + = 0 x t For a rectangular channel : A = bh h h v + v + h = 0 t x x 7
Approximations and solutions Steady state solution Kinematic wave Diffusive wave Full equations 8
Steady state solution (rectangular channel) dv v = g( I I ) g S E dx dh dv v + h = 0 dx dx dh dx 9
Steady state solution (rectangular channel) Uniform flow (no advective acceleration): ( I I ) = 0 S Full solution (insert second equation into first): E dh IS IE v = = 0 with Fr = dx 1 Fr gh yields water surface profiles 10
Classification of profiles h gr = h cr = water depth at critical flow h N = water depth at normal flow I 0 = slope of channel bottom I gr = I cr = critical slope h gr = h cr I gr = I cr Horizontal channel bottom I 0 = 0 H : h > h gr H 3 : h < h gr 11
Classification of profiles Mild slope: h N > h gr I 0 < I gr M1: h N <h > h gr M: h N > h > h gr M3: h N > h < h gr h gr = h cr I gr = I cr Steep slope: h N < h gr I 0 > I gr S1: h N <h > h gr S: h N < h < h gr S3: h N > h < h gr 1
Classification of profiles Critical slope h N = h gr I 0 = I gr C1: h N < h C3: h N > h h gr = h cr I gr = I cr Negative slope I 0 < 0 N: h > h gr N3: h < h gr 13
Examples for composite slopes and transitions h gr = h cr I gr = I cr 14
Numerical solution (explicit FD method) Subcritical flow: Computation in upstream direction hx ( +Δx) hx ( ) IS IE( x+δx) = = Δx 1 Fr( x+δx) 0 Solve for h(x) Supercritical flow: Computation in downstream direction hx ( +Δx) hx ( ) Δx IS IE( x) = = 1 Fr( x) 0 Solve for h(x+ x) 15
Approximations Kinematic wave Diffusive wave Dynamic wave (Full equations) v v h = v g + g IS I t x x h h v + v + h = 0 t x x ( ) E Rectangular channel 16
Approximations In the different approximations different terms in the equation of motion are neglected against the term gi S : 1. Approximation: Kinematic wave g I ( I ) = 0. Approximation: Diffusive wave h g + g IS IE = x Complete solution: Dynamic wave S E ( ) 0 v v h + v + g g IS IE = t x x ( ) 0 17
Kinematic wave Normal flow depth. Energy slope is equal to channel bottom slope. Therefore Q is only a function of water depth. E.g. using the Strickler/Manning equation: Q dq h = x dh x Q= Ak R I = Q( h) /3 1/ str hy S Inserting into the continuity equation yields h h + c ' = 0 t x This is the form of a wave equation (see pressure surge) with wave velocity c = v+c dq dh c ' = b Instead of using Q=Q(h) the equation can be derived using v=v(h) 18
Kinematic wave With the Strickler/Manning equation we get: /3 hb Q = hbkstr I h+ b 1/ S and /3 1/3 1/ 1/ str S str S dq dh bh bh b c' = = k I + k hi b b+ h 3 b+ h ( b+ h) For a broad channel the hydraulic radius is approximately equal to the water depth. The wave velocity then becomes c' = c = v + c 3 v = 5 3 k str I 1/ S h /3 = 5 3 v = v + 3 v 19
Kinematic wave The wave velocity is not constant as v is a function of water depth h. Varying velocities for different water depth lead to self-sharpening of wave front Pressure propagates faster than the average flow. Advantage of approximation: PDE of first order, only one upstream boundary condition required. Disadvantage of approximation: Not applicable for bottom slope 0. No backwater feasible as there is no downstream boundary condition. 0
Diffusive wave Now Q is not only a function of h but also of Insertion into the continuity equation yields: h x = I S I E with I E = I E (Q/A) from Strickler or Darcy-Weisbach h / x h 1 Q + = 0 t b x h 1 Q( h, h/ x) h 1 Q( h, h/ x) h + + = t b h x b ( h/ x) x 0 1
Diffusive wave This equation has the form of an advectiondiffusion equation with a wave velocity w and a diffusion coefficient D: with 0 ' = + x h D x h c t h b h Q c = / ' b x h Q D ) / ( / =
Diffusive wave Using the Strickler/Manning equation and assuming a broad rectangular channel (h = R hy ) one obtains: and 5/3 Q= vhb= kstrbh IS h/ x Insertion into the continuity equation yields with = bv x 3 x I h/ x x Q 5 h vbh h h t c'= + c 5 3 v h x ' D and ( ) S h x D = = 0 vh / ( I h x) S 3
Diffusive wave D is always positive, as the energy slope I = I h/ x E is always positive in flow direction. S The wave moves downstream and flattens out diffusively. A lower boundary condition is necessary because of the second derivative. This allows the implementation of a backwater effect. 4
St. Venant equation as wave equation v v h + v = g + g I I S t x x h h v + v + h = t x x 0 ( ) E Linear combinations: Multiply second equation with ± l and add to first equation 5
St. Venant equation as wave equation v v v h h h + v + λh + λ + λv + g g IS IE = t x x t x x Write derivatives of h and v as total derivatives along a characteristic line: ( ) 0 v v h g h ( v λh) λ ( v ) + + + + + g( I ) S IE = 0 t x t λ x Choosing λ =± g h the two characteristics have the same relative wave velocity (with respect to average water velocity). 6
St. Venant equation as wave equation v v g h h + ( v± gh) ± + ( v± gh) g( IS IE) = 0 t x gh t x The characteristics are therefore: dx = c' = v ± gh = v ± c dt and the relative wave velocity for shallow water waves is c= gh Difference to the surge in pipes: v cannot be neglected in comparison to c 7
General form of the St.-Venant equation General cross section: Width at water surface b(h), cross sectional area A(h) and da(h)/dh = b(h) v v h + v = g + g( I ) S IE t x x h h v bh ( ) + vbh ( ) + Ah ( ) = 0 t x x Rel. wave velocity: c= ga()/() h b h 8
Solution of the full St. Venant equation (Form with b = const. for clarity) v v h = v g + g I I S t x x h h v + v + h = t x x 0 ( ) E The solution is found numerically. E.g. with a difference method. In the following an explicit difference method is discussed first. 9