Dimensional Analysis and Similitude- M3 Part 1(HHM).pptx

Dimensional analysis uses the Buckingham π theorem to create dimensionless parameters, simplifying physical problems. Similitude involves creating scaled models for experimental study, ensuring geometric, kinematic, and dynamic similarity. Applications include fluid mechanics, heat transfer, and structural engineering, with practical examples demonstrating successful model testing and prototyping

1. Dimensional Analysis and Hydraulic Similitude Unit -3

2. Dr. Indrajeet Sahu Module-2 (HHM) VCE 2 Hydraulics/Fluid mechanics is an experimental science & is also a complex subject. It is usually impossible to determine all the essential facts for a given fluid flow by pure theory, &hence, dependence must often be placed up on experimental investigations. The number of tests to be made can be greatly reduced by the systematic use of Dimensional Analysis and the laws of similitude or similarity. There are many cases particularly where localized flow patterns can not be mathematically modelled ,when physical models ( Dimensional analysis ) are still needed. There fore, without the technique of dimensional analysis in experimental and computational progress in fluid mechanics would have been considerably retarded

3. Dimensional analysis Dr. Indrajeet Sahu Module-2 (HHM) VCE 3 Dimensional Analysis is a mathematical technique, which makes use of the study of dimension as an aid to the solution of several engineering problems. Each physical phenomenon can be Expressed by an equation giving relationship between different quantities, such quantities are dimensional and non dimensional. Dimensional analysis helps in determining a systematic arrangement of the variables in the physical relationship, combining dimensional variables to form non-dimensional parameters. It is based on the principle of dimensional homogeneity and uses the dimensions of relevant variables affecting the phenomenon. Dimensional analysis has become an important tool for analyzing fluid flow problems . It is especially useful in representing experimental results in a concise form.

4. Dr. Indrajeet Sahu Module-2 (HHM) VCE 4 Dimensional analysis also forms the basis for the design and operation of physical scale models, which are used to predict the behavior of their full - sized counter parts called 'prototypes ‘. Such models, which are generally geometrically similar to the prototype, are used in the design of aircraft, ships, submarines, pumps, turbines, harbors, breakwaters, river and estuary engineering works, spillways, etc . Working principles of dimensional analysis First predicts the physical parameters that will influence the flow, Then grouping the parameters in dimensionless groups. Application of dimensional analysis: To test the dimensional homogeneity of any equation of fluid motion Developing equations - reducing number of variables in an experiment. Producing dimensionless parameters - establish the principle of model design. Converting one system of units into another. To determine the dimension and unit of a physical quantity in an equation. To establish principles for hydraulics similitude for model study.

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34. Dr. Indrajeet Sahu Module-2 (HHM) VCE 34 Model Analysis Model Analysis is actually an experimental method of finding solutions of complex flow problems . The followings are the advantages of the model analysis. The performance of the hydraulic structure can be predicted in advance from its model. Using dimensional analysis, a relationship between the variables influencing a flow problem is obtained which help in conducting tests. The merits of alternative design can be predicted with the help of model analysis to adopt most economical, and safe design. Note: Test performed on models can be utilized for obtaining, in advance, useful information about the performance of the prototype only if a complete similarity exits between the model and the prototype.

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38. Dr. Indrajeet Sahu Module-2 (HHM) VCE 38 Similitude-Type of Similarities Similitude: is defined as similarity between the model and prototype in every respect, which mean model and prototype have similar properties or model and prototype are completely similar. Three types of similarities must exist between model and prototype. Geometric Similarity Kinematic Similarity Dynamic Similarity

39. Before discussing the technique of dimensional analysis, we first explain the underlying concept of dimensional analysis—the principle of similarity . There are three necessary conditions for complete similarity between a model and a prototype. The first condition is geometric similarity —the model must be the same shape as the prototype, but may be scaled by some constant scale factor. The second condition is kinematic similarity , which means that the velocity at any point in the model flow must be proportional (by a constant scale factor) to the velocity at the corresponding point in the prototype flow Dimensional analysis and similarity Fig.: Kinematic similarity is achieved when, at all locations, the velocity in the model flow is proportional to that at corresponding locations in the prototype flow, and points in the same direction. In other words, ratio of velocity must remain constant.

40. The third and most restrictive similarity condition is that of dynamic similarity . Dynamic similarity is achieved when all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow (force-scale equivalence). As with geometric and kinematic similarity, the scale factor for forces can be less than, equal to, or greater than one. Dimensional analysis and similarity All three similarity conditions must exist for complete similarity to be ensured . In a general flow field, complete similarity between a model and prototype is achieved only when there is geometric, kinematic, and dynamic similarity.

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53. Wind Tunnel Testing We match the Reynolds numbers for the full scale model and prototype. which can be solved for the required wind tunnel speed for the model tests V m For many objects, the drag coefficient levels off at Reynolds numbers above some threshold value. This fortunate situation is called Reynolds number independence. It enables us to extrapolate to prototype Reynolds numbers that are outside of the range of our experimental facility. While drag coefficient C D is a strong function of the Reynolds number at low values of Re, C D often levels off for Re above some value. In other words, for flow over many objects, especially “bluff” objects like trucks, buildings, etc., the flow is Reynolds number independent above some threshold value of Re (Fig., typically when the boundary layer and the wake are both fully turbulent.

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