Laboratoire de Mécanique des Fluides et d’Acoustique LMFA UMR CNRS 5509

ECL - UE-FLE - September 2016

Fluid dynamics and energy Christophe Bailly, Julian Scott, Mikhael Gorokhovski & Lionel Soulhac

Ecole Centrale de Lyon & LMFA - UMR CNRS 5509

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x Course organization q Outline of the course General introduction Chapter 1 : Kinematic properties and fundamental laws (2h) Chapter 2 : Newtonian viscous fluid flow (2h) Chapter 3 : Dimensional analysis - Reynolds number (2h) Chapter 4 : Regimes and flow structures as a function of the Reynolds number (2h) Chapter 5 : Turbulent flows (2h) Chapter 6 : Vorticity and basis of aerodynamics (3h) Chapter 7 : Energy, thermodynamics and compressible flows (3h) Chapter 8 : Heat transfer (2.5h) Chapter 9 : Mixing of fluids (2.5h) Chapter 10 : Combustion and flame (3h)

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ECL UE FLE - September 2016 - cb1

1 - Kinematic properties and fundamental laws

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1 - Kinematic properties and fundamental laws Description of fluid motion

Forces applied to a domain

Microscopic and macroscopic scales Macroscopic quantities Material domain Fluid particle Streamlines and pathlines Reynolds transport theorem Incompressibility condition

Surface and volume forces Stress tensor Conservation of momentum Integral and local formulations Application : rocket motion

Conservation of mass Integral and local formulations Reformulation of the Reynolds theorem

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x Description of fluid motion q Microscopic and macroscopic scales free mean path l

macroscopic scale L

molecule size

view at µm

∼ 10−10

∼ 10−6

length scale air ≃ 6 × 10−8 water ≃ 3 × 10−10

(in meters)

2.7 × 107 molecules in 1µm3 (air) 3.3 × 1010 molecules in 1µm3 (water)

Even the smallest structures (eddies) contained in a boundary layer are large with respect to the microscopic scale. The continuous medium can be characterized by its Knudsen number Kn, Kn = l/L ≪ 1 Avogadro’s number, NA = 6.022 × 1023 molecules in 1 mole Normal conditions T0 = 0o C and P0 = 101325 Pa p For a diatomic molecule, l = γπ/2 (ν/c) 33

ECL UE FLE - September 2016 - cb1

x Description of fluid motion q Macroscopic quantities Let us consider a small fluid volume V of length size d such as l ≪ d ≪ L. The macroscopic properties are determined by averaging over all the molecules contained in V . Hence, density

ρ=

m V

(kg/m3)

∑α mα uα macroscopic velocity U = (m.s−1) m where mα and uα are the mass and velocity of species α m = ∑α mα is the total mass of the volume V These variables are well defined since the number of molecules is huge : the macroscopic description used in fluid dynamics is based on this assumption. Macroscopic quantities are a function of time t and position x, and the macroscopic length scale L is associated with the space variations of these quantities. 34

ECL UE FLE - September 2016 - cb1

x Description of fluid motion q Material domain The flow is described by its velocity U ( x, t), its density ρ( x, t) and by an arbitrary quantity ϕ( x, t) A material point x(t) is defined as a point moving with the fluid, that is dx/dt = U, and a material domain Dm is a set of material points moving with the fluid (it always contains the same fluid particles)

Time-streak marker technique (hydrogen bubbles are used as tracers) applied to the steady flow in a contraction (Schraub et al., 1965)

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x Description of fluid motion q Material domain (con’t) Given domain D bounded by the surface S(t) moving at an arbitrary velocity W

S(t) D(t)

n

W is the local velocity of a point on S(t), this velocity is only defined on S(t)

W

Material domain if W = U n is the unit normal vector pointing outward from the surface S

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x Description of fluid motion q Fluid particle A fluid particle is an elementary material domain (discretization at the macroscopic scale) The size d of a fluid particle is by definition small (d → 0) wrt the macroscopic scale L

xP

d

The fluid particle is usually associated with a material point x P , and the fluid properties are assumed to be constant in its volume V , U = U ( x P , t) and ρ = ρ( x P , t) for instance. A material domain D is also a set of fluid particles

Mass M of a domain D , defined from the following volume integral M=

Z

D

ρ dV

and mass of a fluid particle, m = ρ( x P , t)V 37

ECL UE FLE - September 2016 - cb1

x Description of fluid motion q Streamlines and pathlines The streamlines of a fluid flow are the (imaginary) curves tangential to the instantaneous velocity at a given time t and at every point, determined by U × dx = 0. It corresponds to an Eulerian description of the flow, x and t are independant variables. Flow past an airfoil NACA 64A015 (water tunnel, Rec = 7 × 103, zero incidence). The flow is laminar, and appears to be unseparated (small separation region near the trailing edge) Werlé (1974) in Van Dyke (1982, Fig. 23)

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x Description of fluid motion q Streamlines and pathlines (con’d) In the Lagrangian description, attention is focused on a particular fluid particle : a and t are independant variables, where a is its initial position x = x( a, t). The trajectory (or particle path) is provided by dx/dt = U ( x, t) with x = a at t = 0 as initial condition. b

x( a, t)

b

x3 O

x2

a, t = 0

particle path

x1

There is a particularly interesting case when the flow is steady, that is U = U ( x). The fluid particles then follow streamlines, which coincide with pathlines. 39

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x Description of fluid motion q Material derivative In writing the differentiation of a flow variable ϕ = ϕ( x, t), and making use of the summation convention ∂ϕ ∂ϕ ∂ϕ dxi = dϕ = dt + dt + dx · ∇ ϕ dϕ = ∂t ∂xi ∂t W

For a point x(t) of a given motion, dx =W dt ∂ϕ dϕ = + W · ∇ϕ dt ∂t

x3 O

x2

x(t)

x1

The material derivative D/Dt is introduced for a material point x = x(t), with by definition dx/dt = U D ∂ ∂ ∂ ≡ + U · ∇ = + Ui Dt ∂t ∂t ∂xi 40

ECL UE FLE - September 2016 - cb1

x Description of fluid motion q Material derivative (con’t) Time variation of a flow variable ϕ = ϕ( x(t), t) in following a material point (along the flow) Dϕ ∂ϕ = + U · ∇ϕ = Dt ∂t

∂ϕ {z } | ∂t

+

local rate of change of ϕ at x

∂ϕ Ui | {z∂xi }

convective rate of change of ϕ induced by U

Notation D/Dt introduced by Stokes (ca. 1845) for the convective or material derivative Sir George Gabriel Stokes (1819-1903)

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x Description of fluid motion q Reynolds transport theorem Given domain D(t) bounded by the surface S(t) moving at an arbitrary velocity W

S(t) D(t)

W is the local velocity of the control surface S(t)

n

n is the outward-pointing unit normal vector W Let us consider the integral quantity of a local quantity (per unit volume) ϕ( x, t) over the domain D(t) Z Φ(t) =

D

ϕ( x, t) dV

Reynolds theorem dΦ = dt

Z

∂ϕ ϕW · n dS dV + ∂t | D(t){z } | S(t) {z }

time variation of ϕ

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Z

(

motion of S

ECL UE FLE - September 2016 - cb1

W =0

fixed domain

W =U

material domain

x Description of fluid motion q Reynolds theorem (con’t) Leibniz’s rule d dt

Z b(t) a(t)

f (ξ, t)dξ =

Z b(t) ∂ f (ξ, t) a(t)

∂t

db da dξ + f (b, t) − f ( a, t) dt dt

The Reynolds (transport) theorem is the 3-D generalization of the well-known Leibniz’s rule for differentiating a onedimensional integral with variable limits (a proof is given in your notes)

Gottfried Wilhelm von Leibniz (1646 - 1716) 43

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x Description of fluid motion q Incompressibility condition Reynolds theorem applied to a material domain (W = U) with ϕ ≡ 1 Φ=

Z

D

ϕ dV = V (t)

dΦ dV = = dt dt

Z

S(t)

(volume of the domain D )

ϕ U · n dS =

Z

D

∇ · U dV (by using the divergence theorem)

and for a fluid particle of volume V = V (t) ( ∇ · U < 0 contraction 1 dV = ∇·U V dt ∇ · U > 0 expansion of the fluid particle The relative rate of volume growth is given by ∇ · U (Euler, 1755) Incompressibility condition ∇ · U = 0 (volume of the fluid particle remaining constant during its motion)

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x Conservation of mass q Fundamental principles – conservation of mass – conservation of momentum (Newton’s second law) – conservation of energy will be introduced in Chapter 7 (compressible flows)

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ECL UE FLE - September 2016 - cb1

x Conservation of mass q In a material domain M=

Z

D

ρ dV

d dM = and by construction, dt dt

Z

D

ρ dV = 0

Application of the Reynolds theorem with ϕ = ρ and W = U d dM = dt dt

Z

Z

∂ρ ρ dV = dV + D ∂t D

Z

S

ρU · n dS =

Z  D

∂ρ + ∇ · (ρU ) ∂t



dV

and by considering the limit for a fluid particle (D → V ), a local expression for the conservation of mass is obtained ∂ρ + ∇ · (ρU ) = 0 ∂t

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∂ρ ∂(ρUi ) + =0 ∂t ∂xi

ECL UE FLE - September 2016 - cb1

(1)

x Conservation of mass q Alternative forms ∂ρ Dρ ∂ρ + ∇ · (ρU ) = + U · ∇ρ + ρ∇ · U = + ρ∇ · U = 0 ∂t ∂t Dt

(2)

When the flow is incompressible, ∇ · U = 0, and the conservation of mass takes the simple form Dρ =0 Dt The density of a fluid particle remains constant, which can be easily interpreted, m cst ρ= = V cst

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(mass conservation) (incompressibility)

ECL UE FLE - September 2016 - cb1

x Conservation of mass q Incompressibility condition (revisited) In the general case, the mass conservation can be written 1 Dρ ∇·U = − ρ Dt

∆ρ ∼ Ma2 (see Chap. 7) ρ

U Ma = c

The incompressiblity condition is satisfied for low Mach number flows, Ma ≤ 0.3

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x Conservation of mass q Reformulation of the Reynolds theorem to take into account the mass conservation equation The Reynolds theorem is now applied for the variable ϕ = ρχ and an arbitrary domain D Z Z Z ∂(ρχ) d ρχ dV = dV + ρχW · n dS dt D ∂t D S After some algebra ∂χ ∂χ ∂χ ∂(ρχ) ∂ρ = ρ + χ = ρ − χ∇ · (ρU ) = ρ + ρU · ∇χ −∇ · (ρUχ) ∂t ∂t ∂t ∂t | ∂t {z } = ρDχ/Dt

Alterative form of the Reynolds theorem d dt

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Z

Z

Dχ ρχ dV = ρ dV + Dt D D

Z

S

ρχ(W − U ) · n dS

ECL UE FLE - September 2016 - cb1

(3)

x Conservation of mass q Reformulation of the Reynolds theorem (con’t) For a material domain, W = U, d dt

Z

D

ρχ dV =

Z

D

ρ

Dχ dV Dt

For a fixed domain, W = 0, d dt

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Z

Z

Dχ ρ ρχ dV = dV − Dt D D

Z

ECL UE FLE - September 2016 - cb1

S

ρχU · n dS

x Forces applied to a domain q Introduction The total force applied to a domain D can be split into a surface contribution Fs and a volume contribution Fv , F = Fs + Fv Fv represents a possible body force acting in bulk, long-range electromagnetic and gravitational forces. For the gravitational attraction, Fv =

Z

D

ρg dV

Surface forces Fs are associated with the microscopic interactions across the surface S T

T ( x, t, n) is the force per unit surface (Pa)

n

Fs = dS

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Z

S

T dS

ECL UE FLE - September 2016 - cb1

x Forces applied to a domain q Surface force T

dFs = T ( x, t, n) dS is the force from matter A to matter B across dS

n

A B

dS

Action-reaction principle (third Newton’s law) T ( x, t, −n) = − T ( x, t, n)

For the pressure† , T = − pn where p( x, t) is the pressure (Pa) Fs = −

Z

S

pn dS

S(t) D(t)

†

fluid at rest or inviscid flow (see Chapter 2)

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n W

ECL UE FLE - September 2016 - cb1

x Forces applied to a domain q Stress tensor x3

dS

e3

T˜i = T ( x, t, ei ) ei unit vector along xi

n = e2 T˜2

e2 e1 x1

x2





σ11 σ12 σ13    σ21 σ22 σ23  = σ σ31 σ32 σ33



σ12   T˜2 =  σ22  σ32

Cauchy relation T ( x, t, n) = σ · n i.e. a linear (and explicit) dependence of T wrt the normal vector n of dS (a proof is provided in Appendix) Stress tensor σij , i-th component of T along the direction j The stress tensor σ is symmetric, σij = σji 53

ECL UE FLE - September 2016 - cb1



x Forces applied to a domain q Stress tensor (con’t) Expression of the surface force Fs Fs =

Z

S

Z

T dS =

σ · n dS = S |

Z

∇ · σ dV D {z }

divergence theorem

For the pressure, T = − pn and σ = − pI Auguste (Louis) Cauchy (1789-1857)

Fs = − 

Z

S

pn dS =

Z

S

σ · n dS



−p 0 0   σ =  0 −p 0  0 0 −p

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x Conservation of momentum q Equation of motion Fundamental principle of dynamics applied to a fluid particle of mass ρV Z

Z

DU = σ · n dS + ρg dV ρV Dt |S {z D } Fs + Fv = F

Acceleration (material derivative of U) DU ∂Ui ∂Ui DUi = + U = j Dt i Dt ∂t ∂x j

Z

Z

S

D

σ · n dS =

Z

D

∇ · σ dV = V ∇ · σ

ρg dV = V ρg

Conservation of momentum DU = ∇ · σ + ρg ρ Dt

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ECL UE FLE - September 2016 - cb1

(4)

x Conservation of momentum q Integral equations of motion Using the Reynolds transport theorem (3) and (4), d dt

Z

D

ρU dV =

Z

S

σ · n dS +

Z

D

ρg dV +

Z

S

ρU (W − U ) · n dS

(fixed domain W = 0, and material domain W = U)

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ECL UE FLE - September 2016 - cb1

(5)

x Application : rocket motion q Thrust of a rocket propelling in vacuum Rocket velocity U f (t), exit velocity Ue (wrt an inertial frame) Uf

Control domain D bounded by Si ∪ Se Conservation of mass, Eq. (3) with χ = 1 and W = U f ˙ = d M dt

Si

Z

D

ρ dV = −

Z

Se

ρ (U − U f ) · n d S

= −ρe Se (Ue − U f ) · ne < 0

M(t) Conservation of momentum : forces Z

Se ne x3 O 57

x2

Ue − U f (local frame)

Fs = − pn dS Si | {z }

force exerted by the solid surface on the gas = − FR

Z

pn dS − e } | S{z = −Se pe ne

x1 ECL UE FLE - September 2016 - cb1

Fv =

Z

D

ρg dV = Mg

x Application : rocket motion q Rocket propelling in vacuum (con’t) Conservation of momentum, Eq. (5) d dt

Z

D

ρU dV = −

Z

Se

ρU (U − U f ) · n dS + Fs + Fv

The two first terms can be rearranged as follows, Z   ˙ e ρU (U − U f ) · n dS = −ρe Se (Ue − U f ) · ne Ue = MU − Se

˙ f = M˙ (Ue − U f ) + MU

d dt

Z

D

ρU dV

d = dt

Z

D

˙ f ρ(U − U f ) dV + MU˙ f + MU

Finally, the conservation of momentum can be written as d ˙ MU f + dt 58

Z

D

˙ (Ue − U f )−Se pe ne − FR + Mg ρ (U − U f ) d V = M

ECL UE FLE - September 2016 - cb1

x Application : rocket motion q Rocket propelling in vacuum (con’t) d MU˙ f + dt

Z

D

˙ (Ue − U f ) − Se pe ne − FR + Mg ρ (U − U f ) d V = M

˙ (Ue − U f ) − Se pe ne The (vacuum) engine thrust of the rocket is defined by TR ≡ M and FR is the force exerted on the engine. By neglecting the second term (variation of momentum inside the engine) and the pressure contribution (small term), and assuming U f = cst ˙ (Ue − U f ) − Mg FR · e3 ≃ − M {z } | >0

Ariane V, M = 780 tons, 2 solid boosters P230 ˙ ≃ 2356 kg.s−1, TR ≃ − MU ˙ e = 6500 kN Ue ≃ 2750 m.s−1, − M

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x 1 - Appendix q Cauchy’s tetrahedron Momentum equation (based on Newton’s second law) for a tetrahedral fluid particle x M

e3 M3 n

DU = ρV Dt

M2 e2 M1 e1

S0 = A( M1, M2, M3 ) S1 = A( M, M2, M3 ) = S cos(n · e1 ) = S n1

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Z

S

T dS +

Z

D

ρg dV

3 DU ρV = T (n)S0 − ∑ T (e j )S j + ρg V Dt j =1

When V → 0, V ∼ ǫ3 and S j ∼ ǫ2, hence T (n) = ∑3j=1 T (e j )n j

=⇒

Ti (n) = Ti (e j )n j ≡ σij n j

Stress tensor σij , i-th component of T along the direction j T = σ·n ECL UE FLE - September 2016 - cb1

x 1 - Appendix q « Frequently Asked Questions » tensor : geometric object ; vector – first order tensor 2nd-order tensor, e.g. the Kronecker delta δij , must be independent of a particular choice of coordinate system (introduced in 1846 by W. R. Hamilton, 1805-1865) Aris, R., 1962, Vectors, tensors and the basic equations of fluid mechanics, Dover Publications, Inc., New York.

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x 1 - Appendix q Noninertial reference frame R′ Absolute velocity U ( x/R ) = Ur ( x′/R′ ) +

dr + ΩR′ /R × x′ |dt {z } x′ /R′ fixed

x′ Ω

x3

R

x2 x1

x3′

x2′

R′

x1′

r x = r + x′

Acceleration a( x/R ) = ar ( x′/R′ ) + ae ( x′/R ) + 2ΩR′ /R × Ur ( x′/R′ ) {z } | Coriolis d2r dΩ ′ ′ ′ ae ( x/R ) = 2 + × x + Ω × ( Ω × x ) | {z } dt dt R′ centrifugal force

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ECL UE FLE - September 2016 - cb1

Gaspard-Gustave de Coriolis (1792-1843)

x 1 - Appendix q Conservation of the angular momentum for a fluid particle x of volume V DU = ρV x × Dt

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Z

S

( x × T ) dS +

Z

ECL UE FLE - September 2016 - cb1

D

ρ( x × g ) dV