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