The velocity gradient at the wall normal to the wall

In summary, the problem is asking to calculate the velocity gradient at a point 0.5 m downstream from the leading edge of a flat plate in a hypersonic flow at Mach 10 and standard sea level conditions, given the local shear stress and gas temperature at the wall. The relevant concept is boundary layers and the solution may require finding the viscosity of air as a function of temperature.
  • #1
dat
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Homework Statement


This problem is from "Fundamentals of aerodynamics" by John D. Anderson, Jr (Fifth edition, page 101):
Consider a flat plate at zero angle of attack in a hypersonic flow at Mach
10 at standard sea level conditions. At a point 0.5 m downstream from the
leading edge, the local shear stress at the wall is 282 N/m^2. The gas
temperature at the wall is equal to standard sea level temperature. At this
point, calculate the velocity gradient at the wall normal to the wall.

Homework Equations


Boundary layers

The Attempt at a Solution


18581812_1111291955643147_3221053711068545939_n.jpg
 
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  • #2
dat said:

Homework Statement


This problem is from "Fundamentals of aerodynamics" by John D. Anderson, Jr (Fifth edition, page 101):
Consider a flat plate at zero angle of attack in a hypersonic flow at Mach
10 at standard sea level conditions. At a point 0.5 m downstream from the
leading edge, the local shear stress at the wall is 282 N/m^2. The gas
temperature at the wall is equal to standard sea level temperature. At this
point, calculate the velocity gradient at the wall normal to the wall.

Homework Equations


Boundary layers

The Attempt at a Solution


View attachment 205740
viscosity of air is a function of temperature. In real life, we are not always given all the data needed to solve a problem, and we need to research some of the information, particularly values of physical properties.
 
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1. What is the velocity gradient at the wall normal to the wall?

The velocity gradient at the wall normal to the wall is a measure of how the velocity of a fluid changes as it moves towards and away from a wall. It describes the rate of change of velocity with respect to the distance from the wall.

2. Why is the velocity gradient at the wall important?

The velocity gradient at the wall is important because it affects the behavior of fluids near a wall, such as in boundary layers and boundary shear. It also plays a role in heat and mass transfer processes.

3. How is the velocity gradient at the wall calculated?

The velocity gradient at the wall can be calculated using the derivative of the velocity profile at the wall. This can be done using experimental techniques, such as hot-wire anemometry, or by solving the Navier-Stokes equations for a given flow scenario.

4. What factors affect the velocity gradient at the wall?

The velocity gradient at the wall can be affected by several factors, including the viscosity of the fluid, the velocity of the fluid, the distance from the wall, and the roughness of the wall surface. Additionally, the presence of external forces, such as pressure gradients, can also influence the velocity gradient at the wall.

5. How can the velocity gradient at the wall be used in practical applications?

The velocity gradient at the wall is an important parameter in many practical applications, such as in the design of heat exchangers, pumps, and other fluid handling systems. It can also be used to predict the behavior of fluids in pipes, channels, and other flow scenarios, and to optimize the performance of these systems.

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