CU06997 Fluid Dynamics, formulas Henk Massink , 1-2-2013

Definitions (Wikipedia)Fluid mechanics is the study of how fluids move and the forces on them. (Fluids include liquids and gases.) Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion.

Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluid flow: fluids (liquids and gases) in motion. It has several sub disciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion).

The course Fluid Dynamics will mainly cover fluid flow and a bit of fluid statics. In the course we only deal with water (fresh and salt), we don’t deal with other liquids ore with gases. Water is a incompressible fluid.http://en.wikipedia.org/wiki/Fluid_dynamics

ContentsCU06997 Fluid Dynamics, formulas........................................................................................................1

Definitions (Wikipedia).......................................................................................................................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 staticshttp://en.wikipedia.org/wiki/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]

http://en.wikipedia.org/wiki/Pressure_head

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]http://en.wikipedia.org/wiki/Piezometric_head

Velocity Head (3) u2

2∙ g

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

3

Visualisation flow, streamlines, streaklines, streamtubehttp://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines

Flow rate / Discharge [4] Q=u ∙ A

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

In hydrology, the discharge or outflow of a river is the volume of water transported by it in a certain amount of time. Has to do with the outflow of a catchment area.The flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time.

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

Mechanical energy Etotal=Epotential+Ekinetic=m∙ g ∙d+12m ∙u2=constant

E=¿ Energie [J=Nm]Etotalm ∙g

=d+ u2

2g= constan t

m∙ g

d= z+ y=z+ pρ ∙g

[m ]

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]

http://en.wikipedia.org/wiki/Hydraulic_head

4

Bernoulli’s Equation y1+z1+u12

2g= y2+ z2+

u22

2g=H=constant

Modified Bernoulli’s Equation 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]

http://en.wikipedia.org/wiki/Bernoulli's_Principle

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]

5

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, basic ℜ=u . lν

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

l=¿ Length fluid / surface [m] ℜ=¿ Reynolds Number [1]

Reynolds Number, filled pipe ℜ= ρ∙V ∙Dμ

=V ∙ Dν

Reynolds Number, other ℜ=V .4 ∙ Rν

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

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

ℜ>4000 Turbulent flowℜ<2000Laminar flow

http://en.wikipedia.org/wiki/Reynolds_number

http://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity

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 ]

6

Hydraulic radius of a 50% filled pipe R= AP

=

1212

∙

14∙ π ∙ D2

π ∙ D=14∙D [m ]

7

Hydraulic Diameter D=4 ∙ R

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

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

http://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation

Colebrook-White transition formula 1√ λ

=−2 log( ks3,70 ∙D

+ 2,51ℜ ∙√ λ )

λ=¿ Friction coefficient [1]D = Hydraulic Diameter [m]kS = surface roughness [m]

http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae

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=

hfL

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=¿ Hydraulic gradient [1]

C=√ 8gλ Chezy coefficient [m1/2/s]

Manning V=R23 ∙ S0

12

n

Q=1n ∙A53

P23

∙ S012

V=¿ Mean Fluid Velocity [m/s]R=¿ Hydraulic Radius [m]S0=¿ bed slope [1]A=¿ Wetted Area [m2]P=¿ Wetter Perimeter [m]n=¿ Mannings roughness coefficient [s/m1/3]

C=R16

n

Specific energy E s= y+V 2

2g

V=¿ Mean Fluid Velocity [m/s]

y= pρ ∙g

=¿ Pressure Head / water depth [m]

α and β coefficient (caused by velocity distribution) assumed as 1

E s= y+α ∙V 2

2g

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

b2 ∙C2∙ S0 S0=S f

yn = normal 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 [-]

16

S0=¿ bed slope [1]Sf=¿ Hydraulic gradient caused by friction [1]

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

http://en.wikipedia.org/wiki/Froude_number

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

yc13

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

18

Hydraulic structureshttp://en.wikipedia.org/wiki/Weir

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 ) ∙h152

Q = 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 ∙h32

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]

20

Overflows Q=m∙B ∙ H32

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

measured from top crest!!

21