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# Dimensional Analysis and Dynamic Similitude

الكلية كلية الهندسة/المسيب     القسم هندسة الطاقة     المرحلة 2
أستاذ المادة سناء عبد الرزاق جاسم       09/05/2017 19:58:33
Dimensional Analysis and Dynamic Similitude
Dimensional Analysis:
Dimensional analysis is a mathematical technique used to predict physical parameters that influence the flow in fluid mechanics, heat transfer in thermodynamics, and so forth. The analysis involves the fundamental units of dimensions MLT: mass, length, and time. It is helpful in experimental work because it provides a guide to factors that significantly affect the studied phenomena.
Dimensional analysis is commonly used to determine the relationships between several variables, i.e. to find the force as a function of other variables when an exact functional relationship is unknown. Based on understanding of the problem, we assume a certain functional form.
Basically, dimensional analysis is a method for reducing the number and complexity of experimental variables which affect a given physical phenomenon, by using a sort of compacting technique. If a phenomenon depends upon n dimensional variables, dimensional analysis will reduce the problem to only k dimensionless variables, where the reduction
n - k = 1, 2, 3, or 4, depending upon the problem complexity.

Generally n - k equals the number of different dimensions (sometimes called basic or pri- mary or fundamental dimensions) which govern the problem. In fluid mechanics, the four basic dimensions are usually taken to be mass M, length L, time T, and temperature ? or an MLT? system for short. Sometimes one uses an FLT? system, with force F replacing mass.
Although its purpose is to reduce variables and group them in dimensionless form,
dimensional analysis has several side benefits. The first is enormous savings in time
and money. Suppose one knew that the force F on a particular body immersed in a
stream of fluid depended only on the body length L, stream velocity V, fluid density
?, and fluid viscosity µ, that is,
F = f(L, V, ?, µ)
Suppose further that the geometry and flow conditions are so complicated that our integral
theories (Chap. 3) and differential equations (Chap. 4) fail to yield the solution

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