Physical Models and Similarity

Model AnalysisModel Analysis is actually an experimental method of finding solutions of complex flow problems.

The followings are the advantages of the physical modelsThe 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 helps 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.

Model AnalysisModel: is a small scale replica of the actual structure.Prototype: the actual structure or machine.

PrototypeModel

Classification of ModelsUndistorted or True Models: are those which are geometrically similar to prototype or in other words if the scale ratio for linear dimensions of the model and its prototype is same, the models is called undistorted model. The behavior of prototype can be easily predicted from the results of undistorted or true model.Distorted Models: A model is said to be distorted if it is not geometrically similar to its prototype. For distorted models different scale ratios for linear dimension are used. For example, if for the river, both horizontal and vertical scale ratio are taken to be same, then depth of water in the model of river will be very very small which may not be measured accurately.The followings are the advantages of distorted modelsThe vertical dimension of the model can be accurately measuredThe cost of the model can be reducedTurbulent flow in the model can be maintainedThough there are some advantage of distorted models, however the results of such models cannot be directly transferred to prototype.

Similitude-Type of SimilaritiesSimilitude: 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 SimilarityKinematic SimilarityDynamic Similarity

Similitude-Type of SimilaritiesGeometric Similarity: is the similarity of shape. It is said to exist between model and prototype if ratio of all the corresponding linear dimensions in the model and prototype are equal. E.g.

Where: Lp, Bp and Dp are Length, Breadth, and diameter of prototype and Lm, Bm, Dm are Length, Breadth, and diameter of model.Lr= Scale ratio

Note: Models are generally prepared with same scale ratios in every direction. Such a model is called true model. However, sometimes it is not possible to do so and different convenient scales are used in different directions. Such a models is call distorted model

Similitude-Type of SimilaritiesKinematic Similarity: is the similarity of motion. It is said to exist between model and prototype if ratio of velocities and acceleration at the corresponding points in the model and prototype are equal. E.g.

Where: Vp1& Vp2 and ap1 & ap2 are velocity and accelerations at point 1 & 2 in prototype and Vm1& Vm2 and am1 & am2 are velocity and accelerations at point 1 & 2 in model.Vr and ar are the velocity ratio and acceleration ratio

Note: Since velocity and acceleration are vector quantities, hence not only the ratio of magnitude of velocity and acceleration at the corresponding points in model and prototype should be same; but the direction of velocity and acceleration at the corresponding points in model and prototype should also be parallel.

Similitude-Type of SimilaritiesDynamic Similarity: is the similarity of forces. It is said to exist between model and prototype if ratio of forces at the corresponding points in the model and prototype are equal. E.g.

Where: (Fi)p, (Fv)p and (Fg)p are inertia, viscous and gravitational forces in prototype and (Fi)m, (Fv)m and (Fg)m are inertia, viscous and gravitational forces in model.Fr is the Force ratio

Note: The direction of forces at the corresponding points in model and prototype should also be parallel.

Types of forces encountered in fluid PhenomenonInertia Force, Fi: It is equal to product of mass and acceleration in the flowing fluid.

Viscous Force, Fv: It is equal to the product of shear stress due to viscosity and surface area of flow.

Gravity Force, Fg: It is equal to product of mass and acceleration due to gravity.

Pressure Force, Fp: it is equal to product of pressure intensity and cross-sectional area of flowing fluid.

Surface Tension Force, Fs: It is equal to product of surface tension and length of surface of flowing fluid.

Elastic Force, Fe: It is equal to product of elastic stress and area of flowing fluid.

Dimensionless NumbersThese are numbers which are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force.

As this is ratio of once force to other, it will be a dimensionless number. These are also called non-dimensional parameters. The following are most important dimensionless numbers.Reynolds NumberFroudes NumberEulers NumberWebers NumberMachs Number

Dimensionless NumbersReynolds Number, Re: It is the ratio of inertia force to the viscous force of flowing fluid.Froudes Number, Re: It is the ratio of inertia force to the gravity force of flowing fluid.

Dimensionless NumbersEulerss Number, Re: It is the ratio of inertia force to the pressure force of flowing fluid.Webers Number, Re: It is the ratio of inertia force to the surface tension force of flowing fluid.

Dimensionless NumbersMachs Number, Re: It is the ratio of inertia force to the elastic force of flowing fluid.

Model Laws or similarity LawsWe have already read that for dynamic similarity ratio of corresponding forces acting on prototype and model should be equal. i.e Force of inertial comes in play when sum of all other forces is not equal to zero which mean In case all the forces are equally important, the above two equations cannot be satisfied for model analysis

Model Laws or similarity LawsHowever, for practical problems it is seen that one force is most significant compared to other and is called predominant force or most significant force.

Thus for practical problem only the most significant force is considered for dynamic similarity. Hence, models are designed on the basis of ratio of force, which is dominating in the phenomenon.

Finally the laws on which models are designed for dynamic similarity are called models laws or laws of similarity. The followings are these lawsReynolds Model LawFroudes Model LawEulers Model LawWebers Model Lawmachs Model Law

Froudes Model LawIt is based on Froudes number and states that Froudes number for model must be equal to the Froudes number for prototype.Froudes Model Law is used in problems where gravity forces is only dominant to control flow in addition to inertia force. These problems include:Free surface flows such as flow over spillways, weirs, sluices, channels etc.Flow of jet from orifice or nozzleWaves on surface of fluidMotion of fluids with different viscosities over one another

Froudes Model LawThe Various Ratios for Reynoldss Law are obtained as

Froudes Model LawQ. In the model test of a spillway the discharge and velocity of flow over the model were 2 m3/s and 1.5 m/s respectively. Calculate the velocity and discharge over the prototype which is 36 times the model size.Solution: Given that

For ModelDischarge over model, Qm=2 m3/secVelocity over model, Vm = 1.5 m/secLinear Scale ratio, Lr =36

For PrototypeDischarge over prototype, Qp =?Velocity over prototype Vp=?

Numerical Problem:Q. The characteristics of the spillway are to be studied by means of a geometrically similar model constructed to a scale of 1:10. (i) If 28.3 cumecs, is the maximum rate of flow in prototype, what will be the corresponding flow in model?(i) If 2.4m/sec, 50mm and 3.5 Nm are values of velocity at a point on the spillway, height of hydraulic jump and energy dissipated per second in model, what will be the corresponding velocity height of hydraulic jump and energy dissipation per second in prototype?Solution: Given thatFor ModelDischarge over model, Qm=?Velocity over model, Vm = 2.4 m/secHeight of hydraulic jump, Hm =50 mmEnergy dissipation per second, Em =3.5 NmLinear Scale ratio, Lr =10For PrototypeDischarge over prototype , Qp=28.3 m3/secVelocity over prototype, Vp =?Height of hydraulic jump, Hp =?Energy dissipation per second, Ep =?

Froudes Model Law

Classification of ModelsScale Ratios for Distorted Models

Distorted modelQ. The discharge through a weir is 1.5 m3/s. Find the discharge through the model of weir if the horizontal dimensions of the model=1/50 the horizontal dimension of prototype and vertical dimension of model =1/10 the vertical dimension of prototype.

Distorted modelQ. A river model is to be constructed to a vertical scale of 1:50 and a horizontal of 1:200. At the design flood discharge of 450m3/sec, the average width and depth of flow are 60m and 4.2m respectively. Determine the corresponding discharge in model and check the Reynolds Number of the model flow.

Distorted model