CU06997 Fluid Dynamics / CU03287 Waterstroming: formulas Henk Massink , 31-1-2014

ContentsCU06997 Fluid Dynamics / CU03287 Waterstroming: formulas.............................................................1

Principal symbols / units........................................................................................................................2

Fluid statics.............................................................................................................................................3

Visualisation flow, streamlines, streaklines, streamtube.......................................................................4

Total Head or Bernoulli’s Equation.........................................................................................................4

Turbulent and Laminar flow, Reynolds Number.....................................................................................6

Laminar flow in pipes and closed conduits.............................................................................................7

Turbulent flow in pipes and closed conduits..........................................................................................7

Frictional head losses........................................................................................................................7

Local head losses................................................................................................................................9

Partially full pipes.............................................................................................................................11

Culverts............................................................................................................................................11

Open channel flow...............................................................................................................................13

Frictional head losses, turbulent flow..............................................................................................13

Subcritical and Supercritical flow.....................................................................................................15

Hydraulic structures.........................................................................................................................16

Sewers..................................................................................................................................................17

1

Principal symbols / unitsA=¿ Wetted Area [m2] Natte doorsnede

Cross-sectional area of flowb = width [m] BreedteC = Chézy coefficient [m1/2/s] Coefficient van ChezyC v=¿ velocity coefficient [-] SnelheidscoëfficiëntC c=¿ contraction coefficient [-] Contractiecoëfficiëntd, D = diameter [m] DiameterDm = Hydraulic mean depth [m] Gemiddelde hydraulische diepteE = Energy [J] =[Nm] EnergieEs = specific energy [m] Specifieke energieF = Force [N] KrachtFr = Froude Number [-] Getal van Froudeg = gravitational acceleration [m/s2] ValversnellingH = head [m] Energiehoogtehf , ∆H = frictional head loss [m] Energieverlies tgv wrijvinghL = local head loss [m] Lokaal energieverlieskL = local loss coefficient [-] Lokaal energieverlies coëfficiëntkS = surface roughness [m] WandruwheidL = length [m] Lengtem = mass [kg] Massan = Manning’s roughness coefficient [s/m1/3] Coëfficiënt van manningp* = piezometric pressure [N/m2]= [Pa] Piezometrische drukp = pressure [N/m2] DrukP = wetted perimeter [m] Natte omtrekPs = crest height [m] StuwhoogteQ = discharge, flow rate [m3/s] Debiet, afvoerq = discharge per unit channel width [m3/ms] Debiet per m breedte R, r = radius [m] StraalR = Hydraulic radius [m] Hydraulische straalRe = Reynolds Number [-] Getal van ReynoldsSc = slope of channel bed to give critical flow [-] Bodemverhang voor grenssnelheidSf ,I = slope of hydraulic gradient [-] EnergieverhangS0 = slope of channel bed [-] BodemverhangSs = slope of water surface [-] Drukverhang, verhang wateru,v = velocity [m/s] StroomsnelheidV = mean velocity [m/s] Gemiddelde stroomsnelheidV = volume [m3] Volumeū = average velocity [m/s] Gemiddelde stroomsnelheidy = water depth [m] Waterdiepteyc = critical depth [m] Kritische waterdiepteyn = normal depth [m] Normale waterdieptez = height above datum, potential head [m] Afstandshoogteδ = boundary layer thickness [m] Dikte grenslaagλ = friction factor [m] Wrijvingsfaktorµ = absolute viscosity [kg/ms]=[N s/m2] Absolute viscositeitν = kinematic viscosity [m2/s] Kinematische viscositeitρ = density of liquid [kg/m3] Soortelijk gewichtτ0 = shear stress at solid boundary [N/m2] Schuifspanningξ = (ksie) Loss coefficient [1] Verliescoëfficiëntµ = contraction coefficient [1] Contractiecoëfficiënt

2

Fluid statics

General pressure intensity p= FA

p=¿ Pressure [Pa=N/m2]F=¿ Force [N]A=¿ Area on which the force acts [m2]

Newton Force F=m∙ g

F=¿ Force [N]m=¿ Weight [Kg]g=¿ earths gravity [m/s2]

Fluid Pressure at a point p= FA

= ρ ∙ g ∙ A ∙ yA

=ρ ∙g ∙ y

Pressure Head y= pρ ∙g

p=¿ Pressure [Pa=N/m2]ρ=¿ fluid density [Kg/m3]g=¿ earths gravity [m/s2]y=¿ distance surface to point [m]

ρ freshwater=1000 [Kg/m3]ρ saltwater=1025 [Kg/m3]

Potential Head z

z=¿ height above datum [m]Piezometric Head z1+ y1=z2+ y2

z=¿ height above datum [m]y=¿ distance surface to point [m]Velocity Head u2

2∙ g

u=¿ Fluid Velocity [m/s]g=¿ earths gravity [m/s2]

3

4

Visualisation flow, streamlines, streaklines, streamtube

Flow rate / Discharge Q=u ∙ A

Q=¿ Flow rate [m3/s]u=¿ Fluid Velocity [m/s]A=¿ Wetted Area [m2]

Wetted Area of a filled pipe A=1/4⋅π⋅D2

A=¿ Wetted Area [m2]D=¿ Diameter pipe [m2]

Continuity equation u1 ∙ A1=u2∙ A2=u3∙ A3=constant

(Principle of conservation of mass)

Total Head or Bernoulli’s Equation

Total Head / Energy [m] H= y+z+ u2

2∙ g

y= pρ ∙g

=¿ Pressure Head [m]

z=¿ Potential Head [m]

u❑2

2g=¿ Velocity Head [m]

Bernoulli’s Equation without head loss

y1+z1+u12

2g= y2+ z2+

u22

2g=H=constant

Bernoulli’s Equation with head loss

y1+z1+u12

2g= y2+ z2+

u22

2g+∆H1−2

y= pρ ∙g

=¿ Pressure Head [m]

z=¿ Potential Head [m]u❑2

2g=¿ Velocity Head [m]

∆H=¿ Head Loss [m] g=¿ earths gravity [m/s2]

5

Momentum equation F x=ρ ∙Q (V 2x−V 1 x )

F=¿ Force [N]ρ=¿ fluid density [Kg/m3]Q=¿ Flow rate [m3/s]V 1=¿Mean velocity before [m/s]V 2=¿Mean velocity after [m/s]

Pitot u= 2√2 g ∙h

u=¿ Fluid Velocity [m/s]g=¿ earths gravity [m/s2]h=¿ Difference in pressure [m]

Discharge small orifice Q=C v ∙C c ∙ A0 ∙2√2 g ∙h

Q=¿ Flow rate [m3/s]A=¿ Wetted Area [m2]C v=¿velocity coefficient (0,97-0,99) [-]C c=¿contraction coefficient (0,61-0,66) [-]g=¿ earths gravity [m/s2]h=¿ Difference in pressure [m]

Discharge large orifice Q=23∙ b ∙

2√2g ∙(h232−h132)

Q=¿ Flow rate [m3/s]b=¿ Width orifice [m2]g=¿ earths gravity [m/s2]h1=¿ Difference in pressure from top [m]h2=¿ Difference in pressure from bottom [m]

6

Turbulent and Laminar flow, Reynolds Number

Kinematic viscosity υ= μρ

μ=¿ Absolute viscosity [kg/ms] υ=¿ Kinematic viscosity [m2/s] water, 20°C= 1,00 ∙10−6

ρ=¿ Density of liquid [kg/m3]

Reynolds Number ℜ=V .4 ∙ Rν

υ=¿ Kinematic viscosity [m2/s] water, 20°C= 1,00 ∙10−6

ρ=¿ Density of liquid [kg/m3] V=¿ Mean velocity [m/s]R = Hydraulic Radius = D/4 [m]ℜ=¿ Reynolds Number [1]

ℜ>4000 Turbulent flowℜ<2000Laminar flow

Hydraulic Radius R= AP

R=¿ Hydraulic Radius [m]A=¿ Wetted Area [m2]P=¿ Wetter Perimeter [m]

Hydraulic radius of a filled pipe R= AP

=

14∙ π ∙D2

π ∙D= 14∙ D [m ]

Hydraulic radius of a 50% filled pipe R= AP

=

1212

∙

14∙ π ∙ D2

π ∙ D=14∙D [m ]

Hydraulic Diameter D=4 ∙ R

R=¿ Hydraulic Radius [m]D=¿ Hydraulic Diameter [m]

7

Laminar flow in pipes and closed conduits

Frictional head loss (laminar flow) h f=32 ∙ μ ∙L ∙Vρ ∙g ∙ D2

h f = frictional head loss [m]μ=¿ Absolute viscosity [kg/ms] L=¿ Length between the Head Loss [m]V=¿ mean velocity [m/s]D = Hydraulic Diameter [m]ρ=¿ Density of liquid [kg/m3] g=¿ earths gravity [m/s2]

Wall shear stress (laminar flow) τ0=¿ 4 ∙ μ ∙V

R¿

τ0 = shear stress at solid boundary [N/m2]μ=¿ Absolute viscosity [kg/ms] V=¿ mean velocity [m/s]R=¿ Hydraulic Radius [m]

Turbulent flow in pipes and closed conduits

Head loss / Energy loss ∆H=ξ ∙ u2

2g

∆ H=¿ Head Loss [m]

u2

2g=¿ Velocity Head [m]

ξ=¿ Loss coefficient [1]g=¿ earths gravity [m/s2]

Frictional head losses

Darcy-Weisbach ∆H f= λ∙ LD∙ u

2

2 g=ξ f ∙

u2

2 g

∆ H f=¿ Head Loss due to friction [m] λ=¿ Friction coefficient [1]

u2

2g=¿ Velocity Head [m]

D = Hydraulic Diameter 4R [m]L=¿ Length between the Head Loss [m]g=¿ earths gravity [m/s2]

8

Colebrook-White transition formula 1√ λ

=−2 log( ks

3,70 ∙D+ 2,51

ℜ ∙√ λ )λ=¿ Friction coefficient [1]D = Hydraulic Diameter [m]kS = surface roughness [m]

Colebrook-White and Darcy Weisbach

V=−2√2g ∙ D ∙S f ∙ log( ks3,70 ∙ D

+ 2,51∙ υD√2 g ∙D ∙S f ) with Sf=

hf

L

V=¿ mean velocity [m/s]D = Hydraulic Diameter [m]kS = surface roughness [m]υ=¿ Kinematic viscosity [kg/ms] water, 20°C= 1,00 ∙10−6

Sf = slope of hydraulic gradient [-]hf = frictional head loss [m]L=¿ Length between the Head Loss [m]g=¿ earths gravity [m/s2]

9

Local head losses

Sudden Pipe Enlargement ∆H l=(V 1−V 2)

2

2g

∆ H l=(1−A1A2

)2

∙V 12

2g ξ l=(1−A1A2

)2

∆H l=¿ Head Loss due to sudden pipe enlargement [m] ξ l=¿ Loss coefficient due to sudden pipe enlargement [1]A=¿ Wetted Area [m2]V=¿ Mean Fluid Velocity [m/s]g=¿ earths gravity [m/s2]1= Before enlargement2= After enlargement

Sudden Pipe Contraction ∆H l=(A1A2

−1)2

∙V 22

2gA1≅ 0,6 ∙ A2

∆H l=0,44 ∙V 22

2 g

∆H l=¿ Head Loss due to sudden pipe contraction [m] V❑=¿ Mean Fluid Velocity after sudden pipe contraction

[m/s] g=¿ earths gravity [m/s2]

Tapered Pipe Enlargement ξ t=n ∙(A1A2

−1)2

∆H t=¿ Head Loss due to tapered pipe enlargement [m] ξ s=¿ Loss coefficient due to tapered pipe enlargement [1]A=¿ Wetted Area [m2]1= Before enlargement2= After enlargement

10

n=¿ factor which depends on the widening angel α

11

Submerged Pipe Outlet ∆H o=1∙v12

2g

∆ H o=¿ Head Loss due to submerged pipe outlet [m] v1=¿ Mean Fluid Velocity before pipe outlet [m/s] ξo=1 Loss coefficient due to submerged pipe outlet [1]g=¿ earths gravity [m/s2]

Pipe Bends ∆H b=ξb ∙v2

2g

∆H b=¿ Head Loss due to pipe bend [m] v❑=¿Mean Fluid Velocity [m/s] ξb=¿ Loss coefficient due to pipe bend [1]g=¿ earths gravity [m/s2]

Tabel 4.5 only applies for α = 90o and a smooth pipe.

With α = 90o and a rough pipe, increase ξ with 100%

With α = 45o use 75% ξ900

With α = 22,5o use 50% ξ900

Smooth and rough pipes are explained further on.

12

Partially full pipes

Ap (h )=14∙ D2 ∙arccos (1−2∙ h

D)−( D

2−h)∙ 2√h ∙D−h2

Rp (h )=

14∙ D2∙arccos(1−2 ∙hD )

2√h ∙D−h2−( D2 −h)

Ap=¿ Wetted Area partially filled pipe [m2]Rp=¿ Hydraulic radius partially filled pipe [m]h = water level partially filled pipe [m]D = Diameter pipe [m]

Culverts

Culvert submerged 1ΔΗ tot=ξ tot⋅

vc2

2gξ tot=(ξ i+ξw+ξu+. .. .)

ξ i=( 1μ−1)2

ξw=λ⋅ l4 R

ξu=1

∆H tot=¿ Total Head Loss Culvert [m] ξ tot=¿ Sum of Loss coefficients [1]vc=¿ Mean Fluid Velocity Culvert [m/s] ξ i=¿ Loss coefficient due to contraction [1]ξw=¿ Loss coefficient due to friction [1]ξo=¿ Loss coefficient due to outlet [1]μ=¿ Contraction coefficient [1]g=¿ earths gravity [m/s2]λ=¿ Friction coefficient [1]R=¿ Hydraulic Radius [m]l=¿ Length between the Head Loss [m]

13

Culvert submerged 2 q=m⋅Ac⋅√2g⋅ΔH tot

m= 1√ξ tot

q=¿ Flow rate Culvert [m3/s]m=¿ Discharge coefficient [m]A=¿ Wetted Area Culvert [m2]∆H tot=¿ Total Head Loss Culvert [m] ξ tot=¿ Sum of Loss coefficients [1]g=¿ earths gravity [m/s2]

Culvert partly submerged

Free flow broad crested weir

(Volkomen lange overlaat) h3≤23 H

qv=cv⋅b⋅H32

q=¿ Discharge Culvert [m3/s]b=¿ Width weir [m]cv=discharge coefficient free flow broad crested weir [m1/2/s]H=¿ Head Loss upstream [m] h3=¿ Water level downstream [m]Submerged flow broad crested weir

(Onvolkomen lange overlaat) h3>23 H

qv=col⋅b⋅h3⋅√2 g⋅(H−h3 )

col≈1

√ξ totq=¿ Discharge Culvert [m3/s]b=¿ Width weir [m]col=discharge coefficient submerged flow broad crested weir [1]H=¿ Head Loss upstream [m] h3=¿ Water level downstream [m]Total head (H) and water level (h) measured from crest weir (bed culvert)

14

Open channel flow

V average=QA

=V 1 A1+V 2 A2+V 3 A3

A1+A2+A3

Frictional head losses, turbulent flowMean boundary shear stress τ 0=ρ ∙ g ∙R ∙ S0

τ0 = shear stress at solid boundary [N/m2]R=¿ Hydraulic Radius [m]S0=¿ Slope of channel bed [1]

Chezy V=C ∙√R ∙S f

V=¿ Mean Fluid Velocity [m/s]R=¿ Hydraulic Radius [m]

15

Sf=¿ Slope energy / total head [1]

C=√ 8gλ Chezy coefficient [m1/2/s]

Manning V=R23 ∙ S f

12

n

Q=1n ∙ A

53

P23

∙ S f

12

V=¿ Mean Fluid Velocity [m/s]R=¿ Hydraulic Radius [m]Sf=¿ Slope Total Head [1]A=¿ Wetted Area [m2]P=¿ Wetter Perimeter [m]n=¿ Mannings roughness coefficient [s/m1/3]

C=R16

n

Specific energy H s= y+V 2

2 g

V=¿ Mean Fluid Velocity [m/s]

y= pρ ∙g

=¿ Pressure Head / water depth [m]

Equilibrium / normal depth [m] yn=3√ q2

b2 ∙C2∙ S0 S0=S f

yn = normal depth, equilibrium depth [m]q = discharge [m3/s]b = width [m] S0=¿ bed slope [1]Sf=¿ Hydraulic gradient caused by friction [1]

C=√ 8gλ Chezy coefficient [m1/2/s]

Backwater, direct step method ∆ x=∆ y ∙( 1−Fr2

S0−Sf)

Δx= horizontal distance from point [m]Δy= waterdepth [m]Fr = Froude number [-]S0=¿ bed slope [1]Sf=¿ Hydraulic gradient caused by friction [1]

16

17

Subcritical and Supercritical flow

Critical depth [m] yc=3√ Q2

g ∙ B2

Critical velocity [m/s] V c=2√g ∙ y c

Froude Number Fr= V2√g ∙ yc

= VV c

yc = critical depth [m]Q = discharge [m3/s]B = width [m] Vc = critical velocity [m/s]V = actual velocity [m/s]Fr = Froude number [-]

Subcritical flow Fr < 1 V < Vc

Supercritical flow Fr > 1 V > Vc

Energy loss hydraulic jump ∆ E=( y2− y1)

3

4 ∙ y1∙ y2

ΔH = Energy loss hydraulic jump [m]y1= depth supercritical flow [m]Y2= depth subcritical flow [m]

Critical bed slope Sc=g ∙n2

yc

13

Sc = critical bed slope [-]yc = critical depth [m]n=¿ Mannings roughness coefficient [s/m1/3]

18

Hydraulic structures

Thin plate (sharp crested weirs) Qideal=23∙ b ∙

2√2g ∙ h132

Rehbock formula Q¿ 23∙ 2√2 g ∙(0.602+0.083 ∙ h1Ps

) ∙ b ∙(h1+0.0012)32

30mm<h1<750mm,b>300mm,P s>100mm,h1<P s

Q = discharge [m3/s]b = width [m] h1 = pressure above crest [m]Ps = crest height [m]

Vee weirs Q=Cd ∙815

∙ 2√2g ∙ tan ( θ2 ) ∙h152Q = discharge [m3/s]Cd=¿ discharge coefficient [-] θ=90°, Cd=0.59 h1 = pressure above crest [m]θ = angle vee [°]

Rectangular broad crested weir Qideal=1.705 ∙b ∙ H 1

32

Ackers C f ≅ 0.91+0.21 ∙h1L

+0.24 ∙( h1h1+Ps

−0.35)0.45<

h1L

<0.8 ,0.35<h1

h1+P s<0.6

Q = discharge [m3/s]b = width [m] h1 = pressure above crest [m]Ps = crest height [m]L = length weir [m]Cf = friction coefficient [-]

Discharge broad-crested weir Q=Cd ∙C v ∙23∙ 2√ 2g3 ∙ b ∙h

32

Q = discharge [m3/s]b = width [m] C v=¿velocity coefficient ) [-]Cd=¿ discharge coefficient [-]h = water pressure above crest [m]

19

Venturi flume Q=1.705 ∙ b2 ∙Cd ∙C v ∙ y132

Q = discharge [m3/s]b = width [m] C v=¿velocity coefficient ) [-]Cd=¿ discharge coefficient [-]y1 = pressure above crest [m]

Sewers

Filled pipes ∆ H=L∙ Q2

C2 ∙ R ∙ A❑2

∆H=¿ Head Loss, energy loss [m] Q = discharge pipe [m3/s]L = length of the pipe [m] C=¿ Chezy coefficient [m1/2/s]R=¿ Hydraulic Radius [m]A=¿ Wetted Area, flow surface [m2]

Chezy coefficient C=18 ∙ log [12 Rk ]C=¿ Chezy coefficient [m1/2/s]R=¿ Hydraulic Radius [m]kS = surface roughness [m]

Energy Gradient [-] Sf=i=∆ HL

Sf ,i = slope of hydraulic gradient [-]L = length [m]∆H=¿ Head Loss, energy loss [m]

Overflow / weir sewer Q=m∙B ∙ H32

Q = discharge overflow [m3/s]m = runoff coefficient (1,5 – 1,8) [m1/2/s]B = Width crest overflow [m]

20

H = Head at overflow [m]measured from top crest!!

21