Fluid part, short Flashcards
Someone claims that the absolute pressure in a liquid of constant density doubles when the depth is doubled. Do you agree? Explain,
No, it does not double exactly. The absolute pressure is:
P_abs=P_atm+ρ_liquid gh
Doubling the depth will give double the pressure in the liquid, but the absolute pressure will not double, since the atmospheric pressure remains the same:
P_abs=P_atm+2ρ_liquid gh
The water level of a tank is 20m above ground. A hose is connected to the bottom of the tank, and the nozzle at the end of the hose is pointing straight up. The water surface is open to the atmosphere. What is the maximum height to which the water could rise? What factors would reduce this height?
Without any losses, the maximum height would be 20 m, because there would have to be energy conservation. The pressure just below the nozzle will be the same as the pressure in the bottom of the tank since they have the same height. This pressure is:
ρgh=1000 kg/m^3 *10 m/s^2 *20 m=20 kPa
Using Bernoulli’s equation, which consists of the flow energy, the kinetic energy, and the potential energy:
P_1/ρ+(V_1^2)/2+gz_1=P_2/ρ+(V_2^2)/2+gz_2
Here, it is assumed that the pressure just below the nozzle is the same as the pressure at the bottom of the tank. It is also assumed that the velocity just below the nozzle is very slow compared to right after the nozzle, so it is assumed to be zero. The elevation below the nozzle is the same as at the bottom of the tank – zero. The pressure at the top of the water column is just the atmospheric pressure, the velocity is zero, and we want to find the height at this point:
z_2=(P_1/ρ+(V_1^2)/2+gz_1-P_2/ρ-(V_2^2)/2)/g=((P_1-P_2)/ρ)/g=((200.000 kg/(ms^2 )-101 kg/(ms^2 ))/(1000 kg/m^3 ))/(10 m/s^2 )=19,99 m
Factors that would reduce the height:
Which atmospheric pressure is assumed
The shape of the entrance to the pipe, the shape of the pipe itself, it bends
The inner surface of the pipe
The relationship between pipe diameter and nozzle diameter can influence the velocity
The assumption of V_1=0 is probably off
V_1=0,45 m/s,z_2=20,01 m
What is the difference between fluid and liquid? Explain the concept of Newtonian fluid.
Newtonian fluids are fluids which rates of deformation dβ/dt is linearly proportional to the shear stress τ. The shear stress is:
τ=μ du/dy
Linearly proportional:
τ∝dβ/dt or τ∝du/dy
The difference between solids and fluids is their reaction to shear stress τ. A substance in liquid or gas phase is referred to as a fluid. Difference lies in how well they are able to resist an applied shear stress that tends to change its shape. A solid can resist an applied shear stress by deforming, whereas a fluid continuously deforms under the influence of a shear stress, no matter how small it might be.
In solids, stress is proportional to strain.
In fluids, stress is proportional to strain rate.
When a constant shear force is applied, a solid will eventually stop deforming at some fixed strain angle. A fluid never stops deforming and approaches a constant rate of strain.
What is the momentum-flux correction factor in the momentum analysis of flow systems? For which type(s) of flow is it significant? Explain.
The momentum-flux correction factor is used in Reynold’s Transport Theorem (RTT) to correct for inlets and outlets not being well rounded and therefore not having a uniform velocity. Dimensionless factor.
For uniform flows (apply to all inlets/outlets):
β=1
β is very close to unity (1) for turbulent flows. But it can be quite big for laminar flows.
Explain the concept of boundary layer and what causes a boundary layer to develop.
Due to the no-slip condition, the fluid right next to a stationary surface does not move. This creates a gradient from stationary to a given velocity V, which is the velocity of the fluid outside the boundary layer.
The boundary layer is very thin in the beginning (entrance) and becomes thicker along the inner surface or pipe.
Boundary layer is caused by the viscosity of the fluid, which is the fluid’s ability to withstand deformation. The deformation could come from friction or inertia.
Velocity profile under development in entrance region. Fully developed profile later.
What is the physical significance of the Reynolds number? How is it defined for (a) flow in a circular pipe of inner diameter D and (b) flow in a rectangular duct of cross section a*b?
Momentum over viscous forces. At low Re numbers, the viscous forces are strong enough to hold the fluid in order, and it will be a laminar flow. And the inertial forces (momentum) are small enough to not interrupt it. For high Re numbers, there will be turbulence, because the inertial forces are too big to keep the flow in order, and it will instead be disorderly.
The force that makes the fluid move over the internal force that holds the fluid together and makes it not want to deform.
Re=(inertial forces)/(viscous forces) ⇒ (momentum forces)/(ability to withstand deformation)
(a) Flow in a circular pipe of inner diameter D:
Re=(V_avg D)/v, where v=kinematic viscosity [m^2/ρ]
(b) Flow in a rectangular duct of cross section a*b:
Re=(V_avg D_h)/v, where hydraulic diameter D_h=2ab/(a+b)
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What is cavitation and what causes it?
Cavitation bubbles. Water vaporizes at low temperatures and pressures. Reduces performance. They form in the “cavities” in the liquid.
The pressure becomes so low at e.g., impeller blades, that the saturation temperature becomes low, so that the water vaporizes at lower temperatures. When evaporating bubbles appear on the surface, they are moved away due to pressure differences. They pop, when they are moved away, and all this creates some very high-pressure waves, which can be very destructive to the surface they were on. This can cause failure in the long run. Normal on impeller blades (pumps, fans, turbines and more). Causes vibration and noise.
How does surface roughness affect the pressure drop in a pipe if the flow is turbulent? What would your response be if the flow were laminar?
Moody chart
These equations on the right are valid for both turbulent and laminar flow.
For turbulent flow:
If we have a bigger surface roughness ε, we will get a bigger relative roughness ε/D. For a bigger relative roughness, there will be a bigger friction factor f, which will result in a bigger pressure drop or head loss. There will not be a big chance in velocity from the higher surface roughness, so the Re number will stay approximately the same.
For laminar flow:
If we start in a laminar flow and increase the relative roughness ε/D, the flow will become slower, and therefore have a lower Re number and stay laminar. The friction factor becomes higher, and so does the pressure drop or head loss.
Overall, bigger losses in both cases.
A constant-velocity horizontal water jet from a stationary nozzle hits normally on a vertical flat plate that rides on a nearly frictionless track. As the water jet hits the plate, it begins to move due to the force exerted by the water. Will the acceleration of the plate remain constant or change? Explain.
Assumptions:
The water jet never slows down due to gravity, resistance from the air, pressure loss etc.
No losses, frictionless track.
The water jet has a constant velocity, so therefore its acceleration is zero. When the water hits the plate, its velocity becomes zero instantly, which means that there is a deceleration (negative acceleration) since there is a change in velocity. This deceleration is:
a_water=-V_water/(~0 s)=-~∞ m/s^2
F=ma=~∞
This negative acceleration multiplied by the mass will be the force which pushes the plate forward. At some point, the plate will gain the same velocity as the water, and from there, the velocity will be constant. Therefore, the acceleration of the plate will not remain constant. It will change since it will at some point decrease to zero.
Another explanation is that we know that the plate starts at an acceleration of zero, and from there gains some acceleration from the water. If the acceleration should remain constant, then the velocity of the plate would at some point go towards infinity, which is not possible. It is just not possible for the plate to gain a higher velocity than that of the water.
Explain how the wall shear stress varies along the flow direction for a laminar flow in a pipe. What would change if the flow were turbulent instead?
Turbulent:
Shear stress τ_w is related to the slope of the velocity profile at the surface. When the profile becomes constant (fully developed), the shear stress stays constant as well. Shear stress is highest at the entrance, where the thickness of the boundary layer is the thinnest, and the shear stress decreases until the flow is fully developed.
Laminar:
Since the entrance region is much shorter for laminar flow, the same thing will happen, but faster. Shear stress acts the same for both flows, but it is mostly present in the entrance region, and since the laminar flow has a shorter entrance region, and develops faster, the shear stress will be bigger here than in turbulent flow.
Name different regions in the figure. Explain in your own words what happens in these regions.
Figure of velocity profiles.
Figure.
The red line (cone shaped) is the velocity boundary layer, where the velocity profile is developing. Once we get the fully developed velocity profile (the rightmost velocity profile), we enter the hydrodynamically fully developed region. This means we leave the hydrodynamic entrance region, which lies before this.
Explain the figure and explain/show how to apply it.
Figure of Moody chart.
This is the Moody chart. It shows the correlation between Re number, friction factor, and relative roughness for different flow regimes (laminar, transitional, and turbulent).
The Re number should be known, and based on this, it can be determined which flow regime we are in. If the surface roughness and diameter of the used pipe is known, the relative roughness ε/D can be found. The green line for the closest relative roughness can be followed until the correct Re number is reached. Then, the black lines can be followed to the left to find the correlated friction factor. The friction factor can be used for further calculations, such as calculating the pressure drop or head loss.
Formulate the Bernoulli principle. What type of flow is considered in this law and what approximations apply?
An approximate relation between pressure (flow energy), velocity (kinetic energy), and elevation (potential energy). It can be used for steady, incompressible flow with negligible viscous forces. No dissipation of mechanical energy (conservation of mechanical energy) – no friction that converts mechanical energy into thermal (internal) energy.
P/ρ+V^2/2+gz=constant along a streamline
Since it is constant along a streamline, this means that the equation can be used to find other values about another point on the same streamline:
P_1/ρ+(V_1^2)/2+gz_1=P_2/ρ+(V_2^2)/2+gz_2
When divided by g: Pressure head, velocity head, elevation head. The total head H is constant along a streamline.
Explain and illustrate with examples the concepts of laminar and turbulent flow. How can you distinguish the two mathematically?
Figure of dye injection in laminar and turbulent flow.
Velocity and friction force are key.
Re number describes when a low is in which regime. That is how you can distinguish the two mathematically (where v is the kinematic viscosity):
Re=(V_avg D)/v
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What is flow separation? What causes it? What is the effect of flow separation on the drag coefficient?
Flow separation is when the boundary layer detaches from the surface. The viscous force of the fluid is overcome. Flow separation could be if the flow goes from depending on drag to depending on pressure. That will be the separation point. Separation of the boundary layer. There is a region between them.
Depends on Re number, surface roughness, and the level of fluctuations in the free stream. More separation after the body (where there is a wake with turbulence), when Re number increases.
The separated region has low pressure and low velocity because it is behind a body, in a wake, turbulent flow. For larger separated regions, there is a large pressure drag. More present in laminar flow, which is why golf balls have dimples to up the roughness, decrease laminar flow and thus drag – making the golf ball travel further.
Drag coefficient is parallel to the free stream velocity:
c_D=F_D/(1/2 ρV^2 A)
The drag coefficient becomes lower, as the Re number increases.
