Question

Please answer the most questions you can.

1 Which terms in the Navier-Stokes equations are neglected in the Euler's equation? Discuss briefly the applicability of the Euler's equation. 2 Is Bernoulli equation valid in the boundary layers? Why? 3 In the Buckingham Pi theorem procedure, If we choose 3 repeating parameters (8%) (6%) and in total there are 6 dimensional parameters, then, how many nondimensional (696) parameters will show in the final functional relation f(?) 4 In chapter 7, when we discussed the "incomplete similarity" for the drag force of a surface ship, why the resistance curve obtained (with Froude number) in the model test can not apply directly to the full scale ship? (8%) (8%) 5-For internal laminar pipe flows, how do we define the "entrance length"? 6 Describe the flow characteristics in this "entrance region", of its velocity and pressure changes (8%) 7-what is the Reynolds stress? write the expression and its physical meaning (8%) 8 In a turbulent pipe flow, what do we call the region right next to the wall? In this (896) thin region, is the flow turbulent ? and why?

Answer 1

1. The effects of viscosity is neglected in Eulers equation thus $\tau$ = -pI . The Eulers equation is applicable for compressible flow at high Mach number.thus use of Euler equation allows simulation of flow over the whole aircraft.

2. The Bernoulli equation is not valid for boundary layers due to following assumptions:

i Steady flow

ii No friction

iii Incompressible fluid

iv No work and heat transfer

v Flow along a streamline.

3. Buckingham pi theorem states that if an equation involving k variables is dimensionally homogeneous then it can be reduced to relationship among ( k-r ) where r is minimum no. of reference dimensions. To determine the total no. of non dimension variables we need to follow the below steps :-

STEP I - Mention all the variables given in the problem i.e ( dimension + non- dimensional )  constant. All the variables must be independent in nature to describe the system.

STEP II - Express variables in terms of basic dimension . For fluid mechanics problem basic dimensions will be M,L and T or F,L and T which will be related to newtons second law ( F = m.a ) such that F = MLT^-2

STEP III - Decide required no. of pi terms by using buckingham pi theorem such that no. of pi terms is equal to ( k-r ) where k is no. of variables in problem from STEP I and r is no. of reference dimensions determined from STEP II.

STEP IV - Select variables that can combine to form pi-terms ( repeating variables) . The required no. of repeating variables is equal to required no. of reference dimension.dependent variable should not be chosen as one of repeating variables.

STEP V - pi terms are formed by multiplying one of non repeating variable by product of repeating variables each raised to an exponent that will make combination dimensionless ( e.g x_i , x₁^a , x₂^b _, x₃^c ) where a,b.c are determined so that combination is dimensionless.

STEP VI - repest STEP V for remaining non repeating variables the resulting set of pi terms will correspond to required no. obtain from STEP III

STEP VII - after obtaining required no. of pi terms make sure all pi - terms are dimensionless .

can be checked by simply substituting basic dimension ( M, l, T ) in pi terms

STEP VIII - final form of relationship can be written as

$\pi$₁ = $\phi$ ( $\pi$₂ ,$\pi$₃, .........$\pi$_k-r)

4. As the resistance curve is directly connected with the speed of the ship i.e froude no. so for a full scale ship the power consumption will be more thus there will be production of waveform which will produce interference so the concept cannot be applied directly to full ships.

5.for a laminar flow the entrance length is the function of pipe diameter and dimensionless Reynolds number and Prandtl number.

6. In the entrance region the inviscid upstream flow converges and enters the tube while the pressure drops lineraly.

7. Reynolds stress is component of total stress tensor in a fluid obtained by averaging Navier stokes equation to account for turbulent fluctuations in fluid momentum

Equation : R_ij = -$\tau$_ij / $\rho$

Physical meaning : A changing fluid velocity in x will result in a consequent flux in y or z.

8. The region next to the wall in turbulent flow is viscous sub layer.Yes the flow is turbulent because initially the flow is laminar because the reynolds no. is small but as the fluid crosses a threshold point the flow converts from laminar to turbulent.