FLUIDS EXAM 2 HW6

Lagrangian view of fluid motion

• follows a specific particle through time and space

• D/Dt

Eulerian view of fluid motion

• looks at a fixed point in space that has many different particles moving through it

• d/dt

pathline

shows that location in space that a single particle occupied at all different times in the past

streakline

a snapshot at an instant in time that shows all the different particles that passed through a single point in space at some time in the past

streamline

a line that is a snapshot at an instant in time that is constructed by requiring the line is smooth, continuous, and everywhere tangent to the velocity

Can pathlines cross?

Yes. But
Although 2 particles cannot occupy the same point at the same instant in time they can occupy the same point at different instants of time — thus, pathlines can cross as long as the two particles enter the point at different times.

Can two streaklines cross?

No. They can merge (in a turbulent flow), but once the dyed particles from one streakline origin merge into the location of dyed particles from another streakline origin the two sets of streaklines will encounter the same flow conditions and move together. The merging of the streaklines is "mixing" the dye from two locations and you cannot "unmix" it, which would be required to have the streaklines cross.

Can two streamlines cross?

No. Since streamlines are tangent to the velocity at an instant in time, crossing of streamlines would imply that there are two different velocities at a point at the same time.

In regions where streamlines are close together, what can be said about the velocity relative to places where the streamlines are further apart?

Where streamlines are closer together the velocity magnitude is faster than in places where the streamlines are further apart.

steady turbulent flow

The flow is formally unsteady (i.e., the velocity keeps changing at a point in space), but the average of the velocity at a point is steady, so it can be called “steady in the mean.”

Does a laminar flow have turbulence?

No. By definition a laminar flow has flow along streamlines that can be slowly moving but do not have the chaos of turbulence.

“no-slip” boundary condition

In a real-world viscous flow, the fluid “sticks” to a solid boundary so it has exactly the same velocity of the surface; i.e., at a non-moving boundary the fluid velocity at the boundary is exactly zero.

inviscid approximation

The viscosity of the fluid, and therefore the effects of friction, are ignored.

Does the no-slip boundary condition apply if we use an “inviscid approximation?”

No. The inviscid approximation requires a “free-slip” boundary where the velocity is allowed to slip along the boundary without frictional losses. This is unrealistic, but is sometimes useful.

Why is the “inviscid approximation” useful?

A frictionless flow has the maximum possible velocities and flowrates.

A flow at a large Reynolds Number is [more?, less?] turbulent than a flow at a small

Reynolds number?

more turbulent