FREE SURFACE HYDRODYNAMICS (module 2)

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1 Introduction to FREE SURFACE HYDRODYNAMICS (module 2) Prof Arthur E Mynett Professor of Hydraulic Engineering in addition to the lecture notes by Maskey et al.

2 Introduction to FREE SURFACE HYDRODYNAMICS (module 2) basic principles: (i) physical concepts (ii) mathematical formulations

4 learning objectives: (re)fresh knowledge physical conservation principles: MASS / MOMENTUM / ENERGY mathematical formulations: CONTINUITY equation MOMENTUM equations (F = ma) (Euler, Navier-Stokes) ENERGY equation (Bernouilli)

5 learning objectives: (re)fresh knowledge physical conservation principles: REFERENCE CO-ORDINATE ORDINATE SYSTEM EULER / LAGRANGE description laminar / turbulent flows (500 < Re < 700) mathematical formulations: TOTAL (MATERIAL) DERIVATIVE FROUDE number, REYNOLDS number (Euler, Navier-Stokes) SPECIFIC ENERGY, CRITICAL DEPTH (Bernouilli)

6 basic principles conservation of MASS continuity principle (control volume) for incompressible flow (hydro( hydrodynamics): dynamics): what comes in must go out conservation of MOMENTUM / ENERGY essentially derived from SAME (!) equation: Newton s second Law F = ma

7 example: hydraulic jump given bc s upstream h 1, v 1 => what are downstream h 2, v 2? h 1 -> v 1

8 example: hydraulic jump conservation of MASS continuity principle h 1 v 1 = h 2 v 2 ( what comes in must go out ) conservation of MOMENTUM (mv) F = ma can also be written as Fdt = d(mv) Fdt = ½ g g (h 12 h 22 )dt d(mv) = (mv) out (mv) in = ( v( 2 dth 2 )v 2 ( v 1 dth 1 )v 1

9 excercise: impinging jet given bc s upstream (diameter d 1, velocity v 1 ) => what is horizontal Force on wall F x? F x dt = - (v 1 dt) (¼ d 2 1 ) v 1 d 1 =>v 1 <= F x

10 basic principles (ctd) conservation of MASS continuity principle (control volume) for incompressible flow (hydro( hydrodynamics): dynamics): what comes in must go out conservation of MOMENTUM / ENERGY essentially derived from SAME (!) equation: Newton s second Law F = ma

11 Bernouilli Equation conservation of MASS continuity principle (control volume) for incompressible flow (hydrodynamics): what comes in must go out conservation of MOMENTUM / ENERGY essentially derived from SAME (!) equation: Newton s second Law F = ma

12 Bernoulli Equation basic assumptions steady flow conditions (d/dt=0) valid along a streamline (bottom, free surface, ) practical (hydraulics) formulation z + p/ g g + v 2 /2g = H (constant) [L] z = position head [L] p/ g g = pressure head [L] (z+p/ g) = piezometric head [L] v 2 /2g = velocity head [L] H = energy head [L]

13 example: flow over weir given bc s upstream h1, v1 => what are weir conditions h2, v2? h1 h1 h2, v2 -> v1 / /

14 Bernoulli Equation specific energy (in terms of h, q) (NB appropriate choice of reference frame!!) h + v 2 /2g = H (constant) q = vh leads to h + q 2 /2gh 2 = H (3 (3 rd rd order eq in h)

15 Bernoulli Equation critical depth h c and velocity v c (NB choice of appropriate reference frame!!) h c = 2/3 H v 2 c /2g = 1/3 H viz v 2 c /2g = h c /2 => v 2 c / (gh c ) = Fr 2 = 1

16 Specific Energy (h, q) (ii) super critical flow over weir (NB choice of appropriate reference frame!!) h + v 2 /2g = H q = vh leads to h + q 2 /2gh 2 = H (3 (3 rd rd order eq in h)

17 Energy LOSSES Carnot s Rule sudden expansion (continuity + momentum): H H = H 1 H 2 = (v 1 v 2 ) 2 / 2g

18 example: flow through culvert contraction (coefficient ~ ) expansion (=> energy loss Carnot) pipe flow (=> friction head loss) velocity head (from continuity) piezometric head (from Bernouilli)

19 momentum principle: F = ma a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} } = F i /m u/ t + u/ x dx/dt + u/ u/ y dy/dt + u/ u/ z dz/dt = f x v/ v/ t + v/ v/ x dx/dt + v/ v/ y dy/dt + v/ v/ z dz/dt = f y w/ t + w/ x dx/dt + w/ w/ y dy/dt + w/ w/ z dz/dt = f z

20 momentum principle: F = ma a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} = F i /m u/ t t + u/ x dx/dt + u/ u/ y dy/dt + u/ u/ z dz/dt v/ v/ t t + v/ v/ x dx/dt + v/ v/ y dy/dt + v/ v/ z dz/dt = f x = f y w/ t t + w/ x dx/dt + w/ w/ y dy/dt + w/ w/ z dz/dt = f z

21 momentum principle: F = ma a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} = F i /m u/ t t + u u/ x x + v u/ u/ y + w u/ u/ z v/ v/ t t + u v/ v/ x x + v v/ v/ y + w v/ v/ z w/ t t + u w/ x x + v w/ w/ y + w w/ w/ z = f x = f y = f z D/Dt = TOTAL (material) DERIVATIVE

22 momentum principle: F = mam a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} = F i /m u/ t t + u u/ x x + v u/ u/ y + w u/ u/ z = -1/ 1/ p/ p/ x v/ v/ t t + u v/ v/ x x + v v/ v/ y + w v/ v/ z = -1/ 1/ p/ p/ y w/ t t + u w/ x x + v w/ w/ y + w w/ w/ z = -1/ 1/ p/ p/ z g EULER EQUATIONS

23 momentum principle: F = mam a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} = F i /m Du/Dt = -1/ 1/ p/ p/ x Dv/Dt = -1/ 1/ p/ p/ y Dw/Dt = -1/ 1/ p/ p/ z g EULER EQUATIONS

24 momentum principle: F = mam a = F/m Dv/Dt = F/m D/Dt{v i (t,x,y,z)} = F i /m Du/Dt = -1/ 1/ ( p/ p/ x + xx / x + yx / y + zx / z z ) Dv/Dt = -1/ 1/ ( p/ p/ y + xy / x + yy / y + zy / z ) Dw/Dt = -1/ 1/ ( p/ p/ z z + xz / x + yz / y + zz / z z ) g NAVIER-STOKES EQUATIONS

25 learning objectives: (re)fresh knowledge physical conservation principles: REFERENCE CO-ORDINATE ORDINATE SYSTEM EULER / LAGRANGE description Laminar / turbulent flows (500 < Re < 700) mathematical formulations: TOTAL (MATERIAL) DERIVATIVE FROUDE number, REYNOLDS number (Euler, Navier-Stokes) SPECIFIC ENERGY, CRITICAL DEPTH (Bernouilli)

26 learning objectives: (re)fresh knowledge physical conservation principles: MASS / MOMENTUM / ENERGY mathematical formulations: CONTINUITY equation MOMENTUM equations (F = ma) (Euler, Navier-Stokes) ENERGY equation (Bernouilli)

27 Introduction to FREE SURFACE HYDRODYNAMICS (module 2) basic principles: (i) physical concepts (ii) mathematical formulations

28 Introduction to FREE SURFACE HYDRODYNAMICS (module 2) Dr Shreedhar Maskey Dr Luigia Brandimarte Prof Dano Roelvink

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1 AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01