< 2.9. Substantial derivative > Sensed rates of changeaancl.snu.ac.kr/aancl/lecture/up_file/_1443060174_Chapter2-2(resized).pdf · < 2.11. Pathline, Streamline, and Streakline

Aerodynamics 2008 spring - 1 -

Fundamental Principles & Equations

< 2.9. Substantial derivative >

Sensed rates of change

The rate of change reported by a flow sensor clearly depends on the motion of the sensor. For example, the pressure reported by a static-pressure sensor mounted on an airplane in level flight shows zero rate of change. But a ground pressure sensor reports a nonzero rate as the airplane rapidly flies by a few meters overhead.




Sensed rates of change

Note that although the two sensors measure the same instantaneous static pressure at the same point (at time t=t0), the measured time rates are different.

02

01

0201 tdt

dptdtdpbuttptp




Drifting sensor

We will now imagine a sensor drifting with a fluid element. In effect, the sensor follows the element’s path-line coordinates xs(t), ys(t), zs(t), whose time rates of change are just the local flow velocity components

tzyxudtdztzyxu

dtdytzyxu

dtdx

ssss

ssss

ssss ,,,,,,,,,,,




Drifting sensor

Consider a flow field quantity to be observed by the drifting sensor, such as the static pressure p(x,y,z,t).As the sensor moves through this field, the instantaneous pressure value reported by the sensor is then simply

This ps(t) signal is similar to p2(t) in the example above, but not quite the same, since the p2 sensor moves in a straight line relative to the wing rather than following a path-line like the ps sensor.

ttztytxptp ssss ,,,




Substantial derivative definition

The time rate of change of ps(t) can be computed from above equation using the chain rule.

tp

tz

zp

ty

yp

tx

xp

dtdp ssss





But since dxs/dt etc. are simply the local fluid velocity components, this rate can be expressed using the flow-field properties alone.

The middle expression, conveniently denoted as Dp/Dt in shorthand, is called the substantial derivative of p.

DtDp

zpw

ypv

xpu

tp

dtdps





Although we used the pressure in this example, the substantial derivative can be computed for any flow-field quantity (density, temperature, even velocity)which is a function of x,y,z,t.

V

tzw

yv

xu

tdtD





The rightmost compact D/Dt definition contains two terms.

• The first ∂/∂t term is called the local derivative.

• The second V·∇ term is called the convective derivative.

• In steady flows, ∂/∂t =0, and only the convective derivative contributes.



< 2.10. Fundamental equations >

Recast governing equations

Continuity equation

0

0

0

VDtD

VVt

Vt





Momentum equation

In the same manner,

viscousxx

viscousxx

viscousxx

Fgxp

DtDu

FgxpuV

tuVu

tu

FgxpVu

tu

viscouszz

viscousyy

Fgzp

DtDw

Fgyp

DtDv





Energy equation

viscousviscous WQVfVpqDtVeD

2/2



< 2.11. Pathline, Streamline, and Streakline >

Streamline

A streamline is defined as a line which is everywhere parallel to the local velocity vector V(x,y,z,t)= ui+vj+wk. Define

as an infinitesimal arc-length vector along the streamline.

kdzjdyidxsd ˆˆˆ




Streamline

Since this is parallel to V, we must have

Separately setting each component to zero gives three differential equations which define the streamline.

0ˆˆˆ0

kudyvdxjwdxudzivdzwdy

Vsd




Streamline

In 2-D we have dz=0 and w=0, and only the k component of the equation above is non-trivial. It can be written as an Ordinary Differential Equation for the streamline shape y(x).

uv

dxdy




Pathline

The pathline of a fluid element A is simply the path it takes through space as a function of time.

An example of a pathline is the trajectory taken by one puff of smoke which is carried by the steady or unsteady wind.




Streakline

A streakline is associated with a particular point P in space which has the fluid moving past it.

An example of a streakline is the continuous line of smoke emitted by a chimney at point P, which will have some curved shape if the wind has a time-varying direction.




Streakline

Unlike a pathline, which involves the motion of only one fluid element A in time, a streakline involves the motion of all the fluid elements along its length.

Hence, the trajectory equations for a pathline are applied to all the fluid elements defining the streakline.




Summary

The figure below illustrates streamlines, pathlines, and streaklines for the case of a smoke being continuously emitted by a chimney at point P, in the presence of a shifting wind. One particular smoke puff A is also identified.

In a steady flow, streamlines, pathlines, and streaklines all coincide.



< 2.12. Angular velocity, Vorticity, and Strain >

Fluid element behavior

Consider a moving fluid element which is initially rectangular, as shown in the figure. If the velocity varies significantly across the extent of the element, its corners will not move in unison, and the element will rotate and become distorted.





In general, the edges of the element can undergo some combination of tilting and stretching. For now we will consider only the tilting motions, because this has by far the greatest implications for aerodynamics.





The figure below on the right shows two particular types of element-side tilting motions. If adjacent sides tilt equally and in the same direction, we have pure rotation. If the adjacent sides tilt equally and in opposite directions, we have pure shearing motion.





Both of these motions have strong implications. The absence of rotation will lead to a great simplification in the equations of fluid motion. Shearing together with fluid viscosity produce shear stresses, which are responsible for phenomena like drag and flow separation.




Side tilting analysis

Consider the 2-D element in the xy plane, at time t, and again at time t+Δt.





Points A and B have an x-velocity which differs by ∂u/∂ydy. Over the time interval Δt they will then have a difference in x-displacements equal to

tdyyuxx AB





the associated angle change of side AB is

( assuming small angles )

tyu

dyxx AB

1





A positive angle is defined counterclockwise. We now define a time rate of change of this angle as follows.

yu

tdtd

t

1

0

1 lim

Similar analysis of the angle rate of side AC gives

xv

dtd

2




Angular velocity

The angular velocity of the element, about the z axis in this case, is defined as the average angular velocity of sides AB and AC.

The same analysis in the xz and yz planes will give a 3-D element’s angular velocities ωy and ωx.

yu

xv

dtd

dtd

z 21

21 21

zv

yw

xw

zu

xy 21,

21




Angular velocity

These three angular velocities are the components of the angular velocity vector

kji zyxˆˆˆ

yu

xv

xw

zu

zv

yw

z

y

x

21

21

21




Vorticity

However, since 2ω appears most frequently, it is convenient to define the vorticity vector ξ as simply twice ω.

The components of the vorticity vector are recognized as the definitions of the curl of V, hence we have

kyu

xvj

xw

zui

zv

yw ˆˆˆ2

V




Vorticity

Two types of flow can now be defined :

• Rotational flow. Here ∇×V ≠ 0 at every point in the flow. The fluid elements move and deform, and also rotate.




Vorticity

Two types of flow can now be defined :

• Irrotational flow. Here ∇×V = 0 at every point in the flow. The fluid elements move and deform, but do not rotate.




Strain rate

Using the same element-side angles Δθ1, Δθ2, we can define the strain of the fluid element.

12 strain




Strain rate

This is the same as the strain used in solid mechanics. Here, we are more interested in the strain rate, which is then simply

yu

xv

dtd

dtd

dtstraind

xy

12




Strain rate

Similarly, the strain rates in the yz and zx planes are

xw

zu

zv

yw

zxyz

,

Aerodynamics 2015 fall - 1 -


< 2.13. Circulation >

Circulation

Consider a closed curve C in a velocity field as shown in the figure on the left. The instantaneous circulation around curve C is defined by

C

sdV




Circulation

In 2-D, a line integral is counterclockwise by convention. But aerodynamicists like to define circulation as positive clockwise, hence the minus sign.




Circulation

Circulation is closely linked to the vorticity in the flowfield. By Stokes’s Theorem

A AC

dAndAnVsdV ˆˆ




Circulation

In the 2-D xy plane, we have ξ=ξk and n=k, in which case we have a simpler scalar form of the area integral.

)2( DindAA




Circulation

From this integral one can interpret the vorticity as- circulation per area, or

dAd




Circulation

Irrotational flows, for which ξ=0 by definition, therefore have Γ=0 about any contour inside the flowfield.

Aerodynamic flows are typically of this type.




Circulation

The only restriction on this general principle is that the contour must be reducible to a point while staying inside the flowfield. A contours which contains a lifting airfoil, for example, is not reducible, and will in general have a nonzero circulation.



< 2.14. Stream function >

Definition

Consider defining the components of the 2-D mass flux vector ρV as the partial derivatives of a scalar stream function, denoted by : yx,

xv

yu

,




Definition

For low speed flows, ρ is just a known constant, and it is more convenient to work with a scaled stream function

which then gives the components of the velocity vector V

yx,

xv

yu

,




Example

Suppose we specify the constant-density stream function to be

which corresponds to a vortex flow around the origin.

2222 ln21ln, yxyxyx

2222 ,yx

xx

vyx

yy

u

which has a circular “funnel” shape as shown in the figure. The implied velocity components are then



Streamline interpretation

The stream function can be interpreted in a number of ways. First we determine the differential of ψ as follows.

Now consider a line along which ψ is some constant ψ1.


dxvdyud

dyy

dxx

d

1

1, yx





Along this line, we can state that dψ=dψ1=d(constant)=0, or

which is recognized as the equation for a streamline.

1

uv

dxdydxvdyu 0





Hence, lines of constant ψ(x, y) are streamlines of the flow. Similarly, for the constant-density case, lines of constant ψ(x, y) are streamlines of the flow. In the example above, the streamline defined by

can be seen to be a circle of radius exp(ψ1).

122ln yx




Mass flow interpretation

Consider two streamlines along which ψ has constant values of ψ1 and ψ2. The constant mass flow between these streamlines can be computed by integrating the mass flux along any curve AB spanning them.

1 2




Mass flow interpretation

The mass flow integration proceeds as follows

Hence, the mass flow between any two streamlines is given simply by the difference of their stream function values.

B

A

B

A

B

AddxvdyudAnVm 12ˆ



< 2.15. Velocity potential >

Definition

Consider defining the components of the velocity vector Vas the gradient of a scalar velocity potential function, denoted by φ(x, y, z).

If we set the corresponding x, y, z components equal, we have the equivalent definitions

zk

yj

xiV

ˆˆˆ

zw

yv

xu

,,




Example

Suppose we specify the potential function to be

yxyx arctan,

which has a corkscrew shape as shown in the figure.




Example

The implied velocity components are then

2222 ,yx

xy

vyx

yx

u

which corresponds to a vortex flow around the origin.




Irrotationality

Any velocity field defined in terms of a velocity potential is automatically an irrotational flow. Often the synonymous term potential flow is also used.

0

0,0

022

V

yzzyyzzyzv

yw

zy

x



< 2.16. Stream function & Velocity potential >

Stream function

For a streamline, ψ(x, y) = constant.

Solve above equation for dy/dx,

0

dyudxv

dyy

dxx

d

uv

dxdy

const




Velocity potential

Similarly, for an equipotential line φ(x, y) = constant.

Solve above equation for dy/dx,

0

dyvdxu

dyy

dxx

d

vu

dxdy

const




Relationship

Hence, we have

constconst dxdydxdy

/1

This equation show that the slope of a ψ=constant line is the negative reciprocal of the slope of a φ=constant line, i.e., streamlines and equipotential lines are mutually perpendicular.

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< 2.9. Substantial derivative > Sensed rates of changeaancl.snu.ac.kr/aancl/lecture/up_file/_1443060174_Chapter2-2(resized).pdf · < 2.11. Pathline, Streamline, and Streakline