Differential Analysis of Fluid Flow PDF

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DIFFERENTIAL ANALYSIS OF FLUID FLOW A: Mathematical Formulation (4.1.1, 4.2, 6.1-6.4) B: Inviscid Flow: Euler Equation/Some Basic, Plane Potential Flows (6.5-6.7) C: Viscous Flow: Navier-Stokes Equation (6.8-6.10)

Introduction Differential Analysis z

z

z

z z

There are situations in which the details of the flow are important, e.g., pressure and shear stress variation along the wing…. Therefore, we need to develop relationship that apply at a point or at least in a very small region (infinitesimal volume) with a given flow field. This approach is commonly referred to as differential analysis. The solutions of the equations are rather difficult. Computational Fluid Dynamic (CFD) can be applied to complex flow problems.

PART A Mathematical Formulation (Sections 4.1.1, 4.2, 6.1-6.4)

Fluid Kinematics (4.1.1, 4.2) Kinematics involves position, velocity and acceleration, not forces. z kinematics of the motion: velocity and acceleration of the fluid, and the description and visualization of its motion. z The analysis of the specific force necessary to produce the motion - the dynamics of the motion. z

4.1 The Velocity Field A field representation – representations of fluid parameters as functions of spatial coordinate z

the velocity field

V = u( x , y , z , t )i + v( x , y , z , t ) j + w( x , y , z , t )k r drA uur = VA dt

V = V ( x, y, z, t )

(

V = V = u 2 + v 2 + w2

A change in velocity results in an acceleration which may be due to a change in speed and/or direction.

)

1

2

4.1.1 Eulerian and Lagrangian Flow Descriptions z

Eulerian method: the fluid motion is given by completely prescribing the necessary properties as functions of space and time.

z

From this method, we obtain information about the flow in terms of what happens at fixed points in space as the fluid flows past those points.

z

Lagrangian method: following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time.

V4.3 Cylinder-velocity vectors V4.4 Follow the particles V4.5 Follow the particles

4.1.4 Streamlines, Streaklines and Pathlines z

z

z

A streamline is a line that is everywhere tangent to the velocity field. A streakline consists of all particles in a flow that have previously passed through a common point. A pathline is a line traced out by a given flowing particle.

V4.9 streamlines V4.10 streaklines V4.1 streaklines

4.1.4 Streamlines, Streaklines and Pathlines z

For steady flows, streamlines, streaklines and pathlines all coincide. This is not true for unsteady flows.

z

Unsteady streamlines are difficult to generate experimentally, but easy to draw in numerical computation. On the contrary, streaklines are more of a lab tool than an analytical tool. How can you determine the unsteady pathline of a moving particle?

z

z

4.2 The Acceleration Field z

z

z

The acceleration of a particle is the time rate change of its velocity. For unsteady flows the velocity at a given point in space may vary with time, giving rise to a portion of the fluid acceleration. In addition, a fluid particle may experience an acceleration because its velocity changes as it flows from one point to another in space.

4.2.1 The Material Derivative z

Consider a particle moving along its pathline

uur uur ur uur VA = VA rA , t = VA ⎡⎣ x A ( t ) , y A ( t ) , z A ( t ) , t ⎤⎦

( )

The Material Derivative z

Thus the acceleration of particle A,

uur uur ur uur VA = VA rA , t = V A ⎣⎡ x A (t ) , y A (t ) , z A ( t ) , t ⎦⎤ uur uur uur uur uur ∂V ∂V dx ∂V dy dV aA ( t ) = A = A + A A + A A dt ∂t ∂ x dt ∂ y dt uur ∂VA dz A + ∂z dt uur uur uur uur ∂V ∂V ∂V ∂V = A + u A A + v A A + wA A ∂t ∂x ∂y ∂z

( )

Acceleration z

This is valid for any particle

ur ur ur ur r ∂V ∂V ∂V ∂V a= +u +v +w ∂t ∂x ∂y ∂z ∂u ∂u ∂u ∂u +u +v +w ax = ∂t ∂x ∂y ∂z ∂v ∂v ∂v ∂v = + + + ay u v w ∂t ∂x ∂y ∂z ∂w ∂w ∂w ∂w az = u v w + + + ∂t ∂x ∂y ∂z

Material derivative z

Associated with time variation Acceleration: ur ur ur ur ur ur r DV ∂V ∂V ∂V DV ∂V = +u +v +w a= , ∂t ∂x ∂y ∂z Dt Dt Associated with space variation

z

Total derivative, material derivative or substantial derivative ∂( ) ∂( ) ∂( ) D( ) ∂ ( ) = +u +v +w ∂t ∂x ∂y ∂z Dt r ∂( ) + (V ⋅ ∇ ) ( ) = ∂t

Material derivative z

z

The material derivative of any variable is the rate at which that variable changes with time for a given particle (as seen by one moving along with the fluid – the Lagrangian descriptions) If velocity is known, the time rate change of temperature can be expressed as,

∂T ∂T ∂T DT ∂T = +u +v +w ∂t ∂x ∂y ∂z Dt r ∂T = + (V ⋅ ∇)T ∂t Example: the temperature of a passenger experienced on a train starting from Taipei on 9am and arriving at Kaohsiung on 12.

Acceleration along a streamline ur r ⎛ R3 ⎞ r V = u ( x ) i = V0 ⎜1 + 3 ⎟ i, x ⎠ ⎝

ur ur r ∂V ∂V ⎛ ∂u ∂u ⎞ r a= +u = + u ⎟i ∂t ∂x ⎝⎜ ∂t ∂x ⎠

r r R3 −4 3 a = Vo (1 + 3 )Vo[ R ( −3 x )] i x

4.2.2 Unsteady Effects For steady flow ∂ ( )/∂ t ≡ 0, there is no change in flow parameters at a fixed point in space. For unsteady flow ∂ ( )/∂t ≠ 0. ↓ spatial (convective) derivative ∂T DT DT ∂T v → < 0) = + V ⋅ ∇T (for an unstirred cup of coffee ∂t ∂t Dt Dt ↑ time (local) derivative v v v DV ∂V v = + V ⋅ ∇V ∂t Dt ↑ local acceleration V4.12 Unsteady flow

4.2.3 Convective Effects

DT ∂ T v = + V ⋅ ∇T ∂t Dt ∂T DT = 0 + us ∂s Dt T − Tin =0+u s out Δs

4.2.3 Convective Effects ↓ convective acceleration v v v ∂ DV V v = + V ⋅∇ V Dt ∂t ↑ local acceleration Du ∂u = 0+u Dt ∂x

4.2.4 Streamline Coordinates z

In many flow situations it is convenient to use a coordinate system defined in terms of the streamlines of the flow.

v v V =V s v v Ds v DV DV v a= s +V = Dt Dt Dt ⎛ ∂V ∂V ds ∂V dn ⎞ v =⎜ + + ⎟s ⎝ ∂t ∂s dt ∂n dt ⎠ v v v ⎛ ∂s ∂s ds ∂s dn ⎞ +V ⎜ + + ⎟ ⎝ ∂t ∂s dt ∂n dt ⎠ V4.13 Streamline coordinates

4.2.4 Streamline Coordinates Steady flow

v v ⎛ ∂V ⎞ v ⎛ ∂s ⎞ a = ⎜V ⎟ s + V ⎜V ⎟ s ∂ ⎝ ⎠ ⎝ ∂s ⎠ V2 ∂V v V 2 v ∂V s+ n or as = V , an = =V R R ∂s ∂s v v v v v ⎛ δs δs δ s 1 ∂s δs n ⎞ v = v = δ s , or = , = lim = ⎟⎟ ⎜⎜Q δ s→ 0 δ s δ R s s R s R⎠ ∂ ⎝

6.1 Fluid Element Kinematics z

Types of motion and deformation for a fluid element.

6.1.1 Velocity and Acceleration Fields Revisited z

Velocity field representation ur ur V = V ( x, y, z , t )

z

ur r r r or V = ui + v j + wk

Acceleration of a particle a=

∂V ∂V ∂V ∂V +u +v +w ∂z ∂t ∂x ∂y

ur ur r DV ∂V ur ur = + V ฀∇ V a= Dt ∂ t ∂( ) r ∂( ) r ∂( ) r ∇( ) = j+ k i+ ∂x ∂y ∂z

(

)

ax =

∂u ∂u ∂u ∂u +u +v +w ∂z ∂t ∂x ∂y

ay =

∂v ∂v ∂v ∂v +u +v + w ∂t ∂x ∂y ∂z

az =

∂w ∂w ∂w ∂w +u +v +w ∂z ∂t ∂x ∂y

6.1.2 Linear Motion and Deformation z

variations of the velocity in the direction of ∂u v velocity, ∂x , ∂∂y , ∂w cause a linear stretching ∂z deformation. Consider the x-component deformation:

Linear Motion and Deformation The change in the original volume, δ V = δ xδ yδ z , due to ∂u /∂x : ∂u Change in δ V = ( δ x )(δ yδ z )(δ t ) ∂x Rate change of δ V per unit volume due to ∂u / ∂x : ⎡ ( ∂ u / ∂ x ) δ t ⎤ ∂u 1 d (δ V ) = lim ⎢ ⎥= δ t →0 δ V dt δ t ⎣ ⎦ ∂x

If velocity gradient ∂v / ∂y and ∂w / ∂z are also present, then uv 1 d (δ V ) ∂u ∂v ∂w = + + = ∇ ⋅ V ← volumetric dilatation rate ∂x ∂y ∂z δ V dt z

z

The volume of a fluid may change as the element moves from one location to another in the flow field. For incompressible fluid, the volumetric dilation rate is zero.

6.1.3 Angular Motion and Deformation z

Consider an element under rotation and angular deformation

V6.3 Shear deformation

Angular Motion and Deformation z

the angular velocity of OA is δα ωOA = lim δ t →0 δ t

z

For small angles ∂v δ xδ t ∂v ∂ x = δt tan δα ≈ δα = ∂x δx

so that ⎡ (δ v / δ x )δ t ⎤ δ v ⎥⎦ = δ x δt

ωOA = lim ⎢ δt → 0 ⎣

(if

∂v ∂x

is positive then ω OA will be counterclockwise)

Angular Motion and Deformation z

the angular velocity of the line OB is δβ δ t→ 0 δ t

ωOB = lim

∂u δy δt ∂y ∂u tan δβ ≈ δβ = = δt ∂y δy

so that ⎡ ( ∂u / ∂y ) δ t ⎤ ∂u ⎥= t δ ⎣ ⎦ ∂y

ωOB = lim ⎢ δ t →0

( if

∂u ∂y

is positive, ωOB will be clockwise)

Angular rotation V4.6 Flow past a wing z

The rotation, ω z , of the element about the z axis is defined as the average of the angular velocitiesω OA and ωOB , if counterclockwise is considered to be positive, then, ∂u ⎞

1 ⎛ ∂v

ω z = ⎜⎜ − ⎟⎟ 2 ⎝ ∂ x ∂y ⎠

similarly

ωx =

thus

1 ⎛ ∂u ∂ w ⎞ − ⎟ z ∂ ∂x ⎠ 2⎝

1 ⎛ ∂w ∂v ⎞ ⎜ − ⎟ 2 ⎜⎝ ∂y ∂z ⎟⎠ ,

r

r

r

ωy = ⎜ r

1 2

r

1 2

r

ω = ωx i + ω y j + ω x k = curl V = ∇ ×V

i 1 1 ∂ ∇ ×V = 2 2 ∂x u

j ∂ ∂y v

k ∂ 1 ⎛ ∂w ∂v ⎞ 1 ⎛ ∂ u ∂w ⎞ 1 ⎛ ∂v ∂u ⎞ = ⎜⎜ − ⎟⎟i + ⎜ − ⎟k ⎟ j + ⎜⎜ − 2 ⎝ ∂x ∂y ⎟⎠ ∂z 2 ⎝ ∂y ∂z ⎠ 2 ⎝ ∂z ∂x ⎠ w

Definition of vorticity z

Define vorticity ξ ξ = 2ω = ∇ × V

The fluid element will rotate about z axis as an undeformed block ( ie: ωOA = −ωOB ) only when

∂u ∂v =− ∂x ∂y

Otherwise the rotation will be associated with an angular deformation. z

If

∂u ∂v = ∂y ∂x

or ∇ × V = 0 , then the rotation (and the vorticity )

are zero, and flow fields are termed irrotational.

Different types of angular motions z

Solid body rotation u φ = Ωr

ur = u z = 0

ω z = 2Ω

ωr = ωφ = 0

ωz =

z

1 ∂ 1 ∂u r ruφ ) − ( r ∂r r ∂θ

Free vortex uφ =

k r

ωφ = 0

ωz =

ur = u z = 0 ωr = 0

1 ∂ (ruφ ) = 0 r ∂r

for

r ≠0

Angular Deformation ∂u

z

Apart form rotation associated with these derivatives ∂y ∂v and ∂x , these derivatives can cause the element to undergo an angular deformation, which results in a change in shape of the element.

z

The change in the original right angle is termed the shearing strain δγ , δγ = δα + δβ

where δγ is considered to be positive if the original right angle is decreasing.

Angular Deformation z

Rate of shearing strain or rate of angular deformation ∂u ⎤ ⎡ ∂v t + δ δt ⎢ ∂x δγ y ⎥ ∂ ⎥ γ& = lim = lim ⎢ δ t→0 δ t δ t→0 δt ⎢ ⎥ ⎢⎣ ⎥⎦ ∂v ∂u = + ∂x ∂y

The rate of angular deformation is related to a corresponding shearing stress which causes the fluid element to change in shape. ∂u ∂v =− ∂y ∂x

If , the rate of angular deformation is zero and this condition indicates that the element is simply rotating as an undeformed block.

6.2 Conservation of Mass Conservation of mass:

DM sys Dt

=0

In control volume representation (continuity equation): ∂ ρdV + ∫ ρV • ndA = 0 cs ∂t ∫cv

(6.19)

To obtain the differential form of the continuity equation, Eq. 6.19 is applied to an infinitesimal control volume.

6.2.1 Differential Form of Continuity Equation

∂ ∂ρ ρ dV ≡ δxδyδ z ∫ ∂t ∂t ∂ ρu δx ⎤ ∂ρu δx ⎤ ∂ρ u ⎡ δyδz − ⎡⎢ρu − δy δz = δxδyδz Net mass flow in the direction ⎢ ρ u + ⎥ ⎥ x x x ⎣

∂

2⎦

⎣

∂ρv δxδ yδ z ρy Net mass flow in the z direction ∂ρw δ xδ yδ z ρz ⎡ ∂ρ u ∂ρ v ∂ρ w ⎤ Net rate of mass out of flow ⎢ ∂x + ∂y + ∂ z ⎥ δxδyδz ⎣ ⎦

Net mass flow in the y direction

∂

2⎦

∂

Differential Form of Continuity Equation z

Thus conservation of mass become ∂ρ ∂ρu ∂ρv ∂ρw + + + =0 ∂t ∂x ∂y ∂z

z

(continuity equation )

In vector form r ∂ρ + ∇ ⋅ ρV = 0 ∂t

z

For steady compressible flow ∂ρu ∂ρ v ∂ρw =0 + + ∂z ∂x ∂y

z

r ∇ ⋅ ρV = 0

For incompressible flow ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

r ∇ ⋅V = 0

6.2.2 Cylindrical Polar Coordinates z

The differential form of continuity equation

∂ρ 1 ∂(rρ vr ) 1 ∂ (ρvθ ) ∂ρv z =0 + + + ∂z ∂t r ∂r r ∂θ

6.2.3 The Stream Function z

For 2-D incompressible plane flow then, ∂u ∂v + =0 ∂ x ∂y

z

Define a stream function u=

then z

ψ ( x, y )

such that

∂ψ ∂x

∂ψ ∂y

v=−

∂ ⎛ ∂ψ ⎜ ∂x ⎝ ∂y

⎞ ∂ ⎛ ∂ψ ⎟+ ⎜− ⎠ ∂x ⎝ ∂ y

⎞ ∂ 2ψ ∂ 2ψ − =0 ⎟= x y y x ∂ ∂ ∂ ∂ ⎠

For velocity expressed in forms of the stream function, the conservation of mass will be satisfied.

The Stream Function z

Lines along constant ψ are stream lines Definition of stream line dy = v dx

u

Thus change of ψ , from ( x, y ) to ( x + dx, y + dy) dψ =

∂ψ ∂ψ dx + dy = −vdx + udy ∂x ∂y

Along a line of constant ψ of we have dψ = 0 − vdx + udy = 0

dy v = dx u

which is the defining equation for a streamline. z z

Thus we can use ψ to plot streamline. The actual numerical value of a stream line is not important but the change in the value of ψ is related to the volume flow rate.

The Stream Function Note：Flow never crosses streamline, since by definition the velocity is tangent to the streamlines. z

Volume rate of flow (per unit width perpendicular to the x-y dq = udy − vdx =

∂ψ ∂ψ dy + dx = dψ ∂y ∂x ψ2

q = ∫ dψ = ψ 2 − ψ 1 ψ1

z

If ψ 2 > ψ 1 then q is positive and vice versa. In cylindrical coordinates the incompressible continuity 1 ∂(rvr ) 1 ∂vθ equation becomes, + =0 Then,

vr =

1 ∂ψ r ∂θ

r

∂r

vθ = −

Ex 6.3 Stream function

∂ψ ∂r

r ∂θ

6.3 Conservation of Linear Momentum z

Linear momentum equation F=

or z

∫

CV

r V ρdV +

∫

CS

r r r V ρV ⋅ n dA

Consider a differential system with δm and δV

then z

∑

D V dm ∫ sys Dt r ∂ Fcontents of the = ∂t control volume

δF =

( )

D V δm Dt

Using the system approach then DV = δma δ F = δm Dt

6.3.1 Descriptions of Force Acting on the Differential Element z

Two types of forces need to be considered surface forces：which action the surfaces of the differential element. body forces：which are distributed throughout the element.

z

For simplicity, the only body force considered is the weight of the element, ur ur δ Fb = δ mg

or

δFbx = δmg x

δFby = δmg y

δFbz = δmg z

z

Surface force act on the element as a result of its interaction with its surroundings (the components depend on the area orientation)

Where δFn is normal to the area δA and δF1 and δF2 are parallel to the area and orthogonal to each other.

z

The normal stress

σ nis defined as,

δFn δA→ 0 δA

σ n = lim

and the shearing stresses are define as δF1 δA→ 0 δ A

τ 1 = lim

δF2 δ A→0 δA

τ 2 = lim

we use σ for normal stresses and τ for shear stresses. z

Sign of stresses Positive sign for the stress as positive coordinate direction on the surfaces for which the outward normal is in the positive coordinate direction.

Note：Positive normal stresses are tensile stresses, ie, they tend to stretch the material.

Thus

⎛ ∂σ xx ∂τ yx ∂τ zx ⎞ + + ⎟ δ xδ yδ z ∂ ∂ ∂ x y z ⎝ ⎠

δ Fsx = ⎜

⎛ ∂ τ xy ∂ σ yy ∂ τ zy ⎞ + + ⎟ δ xδ yδ z x y z ∂ ∂ ∂ ⎝ ⎠ ∂τ ⎛ ∂τ ∂σ ⎞ δ Fsz = ⎜ xz + yx + zz ⎟δ xδ yδ z ∂y ∂z ⎠ ⎝ ∂x

δ Fsy = ⎜

ur r r r δ F s = δ Fsx i + δ Fsy j + δ Fsz k ur uur uur δ F = δ Fs + δ Fb

6.3.2 Equation of Motion r r r δFx = δm a x , δFy = δm a y , δFz = δm az δ m = ρδ x δ y δ z Thus ∂ τ yx ∂ σ xx ∂ τ zx ∂u ∂u ∂u ⎞ ⎛ ∂u + + = ρ⎜ + u + v + w ∂x ∂y ∂z ∂x ∂y ∂ z ⎟⎠ ⎝ ∂t ∂ τ xy ∂ σ yy ∂ τ zy ⎛ ∂v ∂v ∂v ∂v ⎞ + + + = ρ ⎜ + u + v + w ⎟ ∂x ∂y ∂z ∂x ∂y ∂z ⎠ ⎝ ∂t

ρgx + ρg

y

ρgz +

∂τ yz ⎛ ∂w ∂ τ xz ∂ σ zz ∂w ∂w ∂w ⎞ + + = ρ⎜ +u +v + w ∂x ∂y ∂z ∂x ∂y ∂ z ⎟⎠ ⎝ ∂t

(6.50)

PART B Inviscid Flow: Euler Equation/Some Basic, Plane Potential Flows (Sections 6.5-6.7)

6.4 Inviscid Flow

6.4.1 Euler’s Equation of Motion z

For an inviscid flow in which the shearing stresses are all zero, and the normal stresses are replaced by -p, thus the equation of motion becomes ⎛ ∂u ∂p ∂u ∂u ∂u ⎞ = ρ⎜ +u +υ +w ⎟ ∂x ∂x ∂y ∂z ⎠ ⎝ ∂t ⎛ ∂υ ∂p ∂υ ∂υ ∂υ ⎞ +υ +w ρgy − = ρ ⎜ + u ∂y ∂x ∂y ∂ z ⎟⎠ ⎝ ∂t

ρgx −

⎛ ∂w ∂p ∂w ∂w ∂w ⎞ =ρ⎜ +u +υ +w ⎟ ∂z ∂x ∂y ∂z ⎠ ⎝ ∂t or uv uv ⎤ ⎡ ∂V uv v ρ g − ∇p = ρ ⎢ + (V ⋅∇ )V ⎥ ⎣ ∂t ⎦

ρgz −

z

The main difficulty in solving the equation is the nonlinear terms which appear in the convective acceleration.

6.4.2 The Bernoulli Equation z

For steady flow uv

uv

ρ g − ∇p = ρ ( V ⋅ ∇ )V v

uv g = − g∇z (up being positive) uv uv 1 uv uv uv uv V ⋅ ∇ V = ∇ V ⋅V −V × ∇ ×V 2

(

)

(

)

(

)

thus the equation can be written as, − ρ g∇ z − ∇ p =

ρ 2

uv uv uv uv ∇ V ⋅ V − ρV × ∇ × V

(

)

(

)

or ∇p

uv uv 1 + ∇ V 2 + g ∇z = V × ∇ × V ρ 2

z

(

( )

)

Take the dot product of each term with a differential length ds along a streamline ∇p

v 1 v v uv uv v ⋅ d s + ∇ V 2 ⋅ d s + g ∇ z ⋅ d s = ⎡V × ∇ × V ⎤ ⋅ d s ⎣ ⎦ ρ 2

( )

(

)

z

uv v Since d s and V are parallel, therefore uv v ⎡V × ( ∇ × V ) ⎤ ⋅ dsv = 0 ⎣ ⎦

z

Since

v v v v d s = dx i + dy j + dz k v ∂p ∂p ∂p ∇p ⋅ d s = dx + dy + dz = dp ∂x ∂y ∂z

Thus the equation becomes dp

( )

1 + d V 2 + g dz = 0 ρ 2

uv

where the change in p, V , and z is along the streamline

z

Equation after integration become dp V 2 ∫ ρ + 2 + gz = constant

which indicates that the sum of the three terms on the left side of the equation must remain a constant along a given streamline. For inviscid, incompressible flow, the equation become, V2 + + gz ...