Wave tank problem: Dimensional Analysis

In summary, the conversation discusses the project of building a scaled wave tank for studying hydrodynamics. The tank will mimic an average wave height of 2.5m, average wave period of 8 seconds, and a seabed depth of 300m. The speaker is seeking guidance on using dimensional analysis to determine the tank and wave dimensions, as they have not done it before. The conversation also mentions the equations and methods involved in dimensional analysis, such as the Buckingham π theorem and the Rayleigh method. An example of using dimensional analysis to solve a problem is provided, which involves the determination of power required to pump water from a depth in a pipe.
  • #1
Whatamiat
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Homework Statement



I am doing a project to build a scaled wave tank.
"This project involves the design, construction and testing of a wave tank.
When studying the hydrodynamics of a scale model or prototype, it is important to choose an appropriate scale.
To determine tank and wave dimensions it is necessary to carry out dimensional analysis."

Tank will have to mimic
Avg. Wave Height 2.5m
Avg. Wave Period 8secs
Seabed depth of 300m

I have room for up to a 10m long x 1m wide tank.

I will have no problem in designing and constructing the tank once I know relevant dimensions.
My problem lies in the using dimensional analysis to determine the dimensions of the tank and waves.
I have not done D.A before and am lost as to where to even start.

2. What to do now

Any guidance of what needs to be done next would be apprieciated.
Do I have all variables I need to determine the dimensions of the tank using D.A?
Can anyone point me to relevant reading material/eqn's/info with regards to this problem and D.A?
 
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  • #2
Homework EquationsThe equations used in dimensional analysis are based on the Buckingham π theorem. This theorem states that for a given set of n variables, with m physical dimensions and n-m independent dimensionless parameters (πi), there is a relationship between the variables that can be expressed as:x1^α1 * x2^α2 * ... * xn^αn = f(π1,π2,...,πn-m) where αi are powers of each variable xi in terms of the m physical dimensions. The unknown powers αi can be determined if the values of the independent dimensionless parameters are known.The most common form of dimensional analysis is the Rayleigh method. This method involves expressing the variables as the product of a characteristic speed and a characteristic length. This has the form:x1 = a1*Vx2 = a2*Lwhere V is the characteristic speed and L is the characteristic length. The characteristic speed and length can then be related to the original variables using the Buckingham π theorem.Solved ProblemsAn example of a problem solved using dimensional analysis is the determination of the power required to pump water from a depth h in a pipe of diameter D. The equation for this problem is:P = ρgQh/D^4where P is the power, ρ is the density of water, g is the acceleration due to gravity, Q is the volumetric flow rate and h is the depth.Using the Rayleigh method, the variables can be expressed as:P = a1*ρV^2Q = a2*LVh = a3*LD = a4*LSubstituting these into the original equation, we get:a1*ρV^2 = (a2*LV)(a3*L)/(a4*L)^4Rearranging this equation gives:V = a5*L^3*ρ^(-1/2)where a5 is a constant.This equation can then be used to calculate the power required to pump water of a given depth in a pipe of a given diameter.
 

1. What is the wave tank problem?

The wave tank problem is a well-known experiment in fluid dynamics that involves creating waves in a tank of water and measuring their properties, such as amplitude, wavelength, and frequency. It is used to study the behavior of water waves and their interactions with different types of boundaries and obstacles.

2. Why is dimensional analysis important in the wave tank problem?

Dimensional analysis is important in the wave tank problem because it allows us to understand the relationship between the different variables involved in the experiment. By using the principles of dimensional analysis, we can identify the key parameters that affect the behavior of the waves and make predictions about how they will change under different conditions.

3. What are the key variables in the wave tank problem?

The key variables in the wave tank problem include the wave properties (such as amplitude, wavelength, and frequency), the properties of the fluid (such as density and viscosity), and the properties of the tank and any boundaries or obstacles present (such as size and shape). These variables are interrelated and can be analyzed using dimensional analysis to understand their impact on the behavior of the waves.

4. How does scaling affect the results of the wave tank problem?

Scaling is an important concept in the wave tank problem because it allows us to simulate real-world conditions in a laboratory setting. By scaling the size and properties of the tank, fluid, and boundaries, we can recreate the behavior of waves in different environments, such as the ocean or a small pond. This allows us to make predictions about how waves will behave in these different scenarios.

5. What are the applications of the wave tank problem?

The wave tank problem has many practical applications, including in the design of ships, offshore structures, and coastal defenses. It is also used in oceanography to study the behavior of waves in the ocean and their impact on coastal areas. Additionally, the principles of dimensional analysis and scaling used in the wave tank problem can be applied to other areas of science and engineering to analyze complex systems and make predictions about their behavior.

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