Dimensional analysis is a mathematical technique which makes the use of the study of the dimension for solving several engineering problems.

• To test the dimensional homogeneity of any equation of fluid motion.

• To derive rational formula for a flow phenomenon.

• To plan model test and present experimental results in a systematic manner.

• To derive equations expressed in terms of non-dimensional parameters to show the relative significance of each parameter.

Dimensional analysis is a mathematical technique which makes the use of the study of the dimension for solving several engineering problems.

• To test the dimensional homogeneity of any equation of fluid motion.

• To derive rational formula for a flow phenomenon.

• To plan model test and present experimental results in a systematic manner.

• To derive equations expressed in terms of non-dimensional parameters to show the relative significance of each parameter.

Based on the Fouriers principle of homogeneity, some of the methods of dimensional analysis are:

• Rayleighs Method

• Buckinghams `pi` Method.

Rayleighs Method:

Rayleigh method of dimensional analysis is a method that gives a special form of relationship among the dimensionless groups, which can be used for determining the expression of a variable which depends upon maximum three or four variables only. This method has an inherent drawback that it does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensionless analysis.

This method employs the following procedures:

• First of all, gather all the independent variables which influence the dependent variable.

• Write down the functional relationship.

• Change the functional relationship to the equation in exponential terms.

• Express each of the quantities (variables) on  both sides of the equation in terms of fundamental dimensions.

• Solve the equation simultaneously and find the exponents.

Let X is a variable , which depends on `X_1,X_2,...X_n` independent variables. Mathematically, it can be written as,

`X=f (X_1,X_2, X_3,...X_n)`...(i)

Now, by Rayleigh method, it can be written as,

`X=K(X_1)^a*(X_2)^b*...(X_n)^z`...(ii)

Where, `K` is a dimensionless constant and the exponents a, b, ..z are found by comparing the power of fundamental dimensions on both sides of the equation and thus the equation is obtained for the dependent variables.

Buckingham `pi` Theorem:

Buckinghams method of dimensional analysis is an improvement over Rayleigh method, which let us know in advance analysis as to how ,any dimensionless groups are to be expected. Buckingham designated these dimensionless group by a greek letter `pi`. It is therefore called as Buckingham`pi` theorem.

It states, �if there are n variables (dependent and independent variables) in a dimensionally homogeneous equation and if these variables contains m fundamental dimensions (such as M, L, T, etc) them the variables are arranged into (n-m) dimensionless terms. These dimensionless terms are called pi terms.

Let any variable `X_1` depends on independent variables `X_2`, `X_3`,...`X_n`. Mathematically, it can be written as,

`X_1=f (X_2, X_3,...X_n)`

or,`F(X_1,X_2, X_3,...X_n)=0`.....(i)

It is a dimensionally homogeneous equations and contains `n` variables. If this equation contains `m` fundamental dimensions, then by buckinghams `pi` theorem,  this equation can be written in terms of `n-m` `pi` terms. Thus,

`F_1(pi_1,pi_2,....pi_(n-m))=0`...(ii)

Where each `pi` terms contains `m+1` variables where `m=3, M L T` are fundamental variables. The `m` variables which appears repeatedly in each of the `pi` terms are called repeating variables.

Selection of Repeating variables:

The following points should be kept in view while selecting `m` repeating variables:

1. The repeating variables must not form the non-dimensional parameters among themselves.

2. As far as possible, the dependent variables should not be selected as repeating variables.

3. No two repeating variables should have the same dimensions.

4. `m` repeating variables must contain jointly all the fundamental dimensions involved in the phenomenon.

5.  The repeating variables should be chosen in such a way that one variable contains geometric property (length, diameter, height), the other variables should contains flow property (velocity, acceleration) and the third variables should contain fluid property (mass density `rho`, weight density `w`, dynamic viscosity `mu`). The choice of repeating variables in most of the fluid mechanics problems may be,

1. `L,V,rho`

2. `D, V, rho`

3. `L, V, mu`

4. `D, V, mu`

Based on the Fouriers principle of homogeneity, some of the methods of dimensional analysis are:

• Rayleighs Method

• Buckinghams `pi` Method.

Rayleighs Method:

Rayleigh method of dimensional analysis is a method that gives a special form of relationship among the dimensionless groups, which can be used for determining the expression of a variable which depends upon maximum three or four variables only. This method has an inherent drawback that it does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensionless analysis.

This method employs the following procedures:

• First of all, gather all the independent variables which influence the dependent variable.

• Write down the functional relationship.

• Change the functional relationship to the equation in exponential terms.

• Express each of the quantities (variables) on  both sides of the equation in terms of fundamental dimensions.

• Solve the equation simultaneously and find the exponents.

Let X is a variable , which depends on `X_1,X_2,...X_n` independent variables. Mathematically, it can be written as,

`X=f (X_1,X_2, X_3,...X_n)`...(i)

Now, by Rayleigh method, it can be written as,

`X=K(X_1)^a*(X_2)^b*...(X_n)^z`...(ii)

Where, `K` is a dimensionless constant and the exponents a, b, ..z are found by comparing the power of fundamental dimensions on both sides of the equation and thus the equation is obtained for the dependent variables.

Buckingham `pi` Theorem:

Buckinghams method of dimensional analysis is an improvement over Rayleigh method, which let us know in advance analysis as to how ,any dimensionless groups are to be expected. Buckingham designated these dimensionless group by a greek letter `pi`. It is therefore called as Buckingham`pi` theorem.

It states, �if there are n variables (dependent and independent variables) in a dimensionally homogeneous equation and if these variables contains m fundamental dimensions (such as M, L, T, etc) them the variables are arranged into (n-m) dimensionless terms. These dimensionless terms are called pi terms.

Let any variable `X_1` depends on independent variables `X_2`, `X_3`,...`X_n`. Mathematically, it can be written as,

`X_1=f (X_2, X_3,...X_n)`

or,`F(X_1,X_2, X_3,...X_n)=0`.....(i)

It is a dimensionally homogeneous equations and contains `n` variables. If this equation contains `m` fundamental dimensions, then by buckinghams `pi` theorem,  this equation can be written in terms of `n-m` `pi` terms. Thus,

`F_1(pi_1,pi_2,....pi_(n-m))=0`...(ii)

Where each `pi` terms contains `m+1` variables where `m=3, M L T` are fundamental variables. The `m` variables which appears repeatedly in each of the `pi` terms are called repeating variables.

Selection of Repeating variables:

The following points should be kept in view while selecting `m` repeating variables:

1. The repeating variables must not form the non-dimensional parameters among themselves.

2. As far as possible, the dependent variables should not be selected as repeating variables.

3. No two repeating variables should have the same dimensions.

4. `m` repeating variables must contain jointly all the fundamental dimensions involved in the phenomenon.

5.  The repeating variables should be chosen in such a way that one variable contains geometric property (length, diameter, height), the other variables should contains flow property (velocity, acceleration) and the third variables should contain fluid property (mass density `rho`, weight density `w`, dynamic viscosity `mu`). The choice of repeating variables in most of the fluid mechanics problems may be,

1. `L,V,rho`

2. `D, V, rho`

3. `L, V, mu`

4. `D, V, mu`

Model analysis is an experimental method of finding solutions of a complex flow problems.

A Model is a small scale replication of the actual machine or structure. The actual machine or structure is called prototype.

Advantages of Model Analysis:

• The performance of hydraulic structures, hydraulic machines can be predicted using model.

• Model testing can be done to detect the defects of an existing structures.

• It is the only means of determining the safety and reliability of a particular portion of a structure in which clear cut analytical and reliable method is not available.

• The model test are quite economical and convenient.

Application of Model Analysis:

• For analysis of civil engineering structures such as dams, spillways, canals, weirs etc.

• Flood control mechanism

• Turbines, pumps and compressor

• Aeroplanes, rocket missiles

• Tall buildings

Types of Models:

The hydraulic models, in general, are classified as:

• Undistorted models

• Distorted models

Undistorted Models:

An undistorted model is the one which is geometrically similar to its prototype. The conditions of similitude are completely satisfied for such models, hence the results obtained from the model tests are easily used to predict the performance of prototype body. In such models, the design and construction of the model and the interpretation of the model results are simpler.

Distorted Models:

A distorted model is the one which is not geometrically similar to its prototype. In such a model, different scale ratios for the linear dimensions are adopted.

A distorted model may have the following distortions:

• Geometrical distortion

• Material distortion

• Distortion of hydraulic quantities

Typical examples for which distorted models are to be prepared are;

• Rivers

• Dams across very wide rivers

• Harbors etc.

Two scale ratios are adopted, horizontal scale ratio for horizontal dimensions, `L_(r_H)` and Vertical scale ratio for vertical dimensions, `L_(r_V)`.

Scale ratio for horizontal dimensions is,

`L_(r_H)=L_p/L_m=B_p/B_m`

Scale ratio for vertical dimensions is,

`L_(r_V)=D_p/D_m` 

Similarly, the scale ratio for velocity based on froude model law is,

`(F_r)_m=(F_r)_p`

`V_m/root(2)D_m=V_p/root(2)D_p`

`V_p/V_m=root(2)(D_p/D_m)=root(2)L_(r_V)`

Similarly, the scale ratio for area of flow is,

`A_p/A_m=(B_p*D_p)/(B_m*D_m)=(L_r)_H*(L_r)_V`

Similarly, the scale ratio for discharge through model and prototype is,

`Q_p/Q_m=(V_p*A_p)/(V_m*A_m)`

`=root(2)(D_p/D_m)*(b_p*D_p)/(b_m*D_m)`

`=b_p/b_m*(D_p/D_m)^(3/2)`

`=(L_r)_H*((L_r)_V)^(3/2)  `

Reasons for adopting distorted Models:

• Maintaining the accuracy in vertical dimensions.

• Maintaining turbulent flow.

• Obtaining suitable roughness conditions.

• Obtaining suitable bed material and its adequate movements.

• Accommodating the available facilities such as money, water supply, space etc.

Merits of distorted models:

• The surface tension can be reduced.

• Model size can be reduced sufficiently.

• The Reynolds number of flow can be increased that will yield better results.

• Accurate measurement can be done.

• Sufficient attractive force can be developed to move the bed material of the model.

Demerits of distorted models:

• The pressure and velocity distribution are not truly reproduced.

• A model wave may differ in type and possibly in action from that of the prototype.

• Slopes of river beds, earthcuts and dikes cannot be truly reproduced.

• It is difficult to extrapolate and interpolate the results obtained from the distorted models.

• The observer may experience an unfavourable psychological effect.

Model analysis is an experimental method of finding solutions of a complex flow problems.

A Model is a small scale replication of the actual machine or structure. The actual machine or structure is called prototype.

Advantages of Model Analysis:

• The performance of hydraulic structures, hydraulic machines can be predicted using model.

• Model testing can be done to detect the defects of an existing structures.

• It is the only means of determining the safety and reliability of a particular portion of a structure in which clear cut analytical and reliable method is not available.

• The model test are quite economical and convenient.

Application of Model Analysis:

• For analysis of civil engineering structures such as dams, spillways, canals, weirs etc.

• Flood control mechanism

• Turbines, pumps and compressor

• Aeroplanes, rocket missiles

• Tall buildings

Types of Models:

The hydraulic models, in general, are classified as:

• Undistorted models

• Distorted models

Undistorted Models:

An undistorted model is the one which is geometrically similar to its prototype. The conditions of similitude are completely satisfied for such models, hence the results obtained from the model tests are easily used to predict the performance of prototype body. In such models, the design and construction of the model and the interpretation of the model results are simpler.

Distorted Models:

A distorted model is the one which is not geometrically similar to its prototype. In such a model, different scale ratios for the linear dimensions are adopted.

A distorted model may have the following distortions:

• Geometrical distortion

• Material distortion

• Distortion of hydraulic quantities

Typical examples for which distorted models are to be prepared are;

• Rivers

• Dams across very wide rivers

• Harbors etc.

Two scale ratios are adopted, horizontal scale ratio for horizontal dimensions, `L_(r_H)` and Vertical scale ratio for vertical dimensions, `L_(r_V)`.

Scale ratio for horizontal dimensions is,

`L_(r_H)=L_p/L_m=B_p/B_m`

Scale ratio for vertical dimensions is,

`L_(r_V)=D_p/D_m` 

Similarly, the scale ratio for velocity based on froude model law is,

`(F_r)_m=(F_r)_p`

`V_m/root(2)D_m=V_p/root(2)D_p`

`V_p/V_m=root(2)(D_p/D_m)=root(2)L_(r_V)`

Similarly, the scale ratio for area of flow is,

`A_p/A_m=(B_p*D_p)/(B_m*D_m)=(L_r)_H*(L_r)_V`

Similarly, the scale ratio for discharge through model and prototype is,

`Q_p/Q_m=(V_p*A_p)/(V_m*A_m)`

`=root(2)(D_p/D_m)*(b_p*D_p)/(b_m*D_m)`

`=b_p/b_m*(D_p/D_m)^(3/2)`

`=(L_r)_H*((L_r)_V)^(3/2)  `

Reasons for adopting distorted Models:

• Maintaining the accuracy in vertical dimensions.

• Maintaining turbulent flow.

• Obtaining suitable roughness conditions.

• Obtaining suitable bed material and its adequate movements.

• Accommodating the available facilities such as money, water supply, space etc.

Merits of distorted models:

• The surface tension can be reduced.

• Model size can be reduced sufficiently.

• The Reynolds number of flow can be increased that will yield better results.

• Accurate measurement can be done.

• Sufficient attractive force can be developed to move the bed material of the model.

Demerits of distorted models:

• The pressure and velocity distribution are not truly reproduced.

• A model wave may differ in type and possibly in action from that of the prototype.

• Slopes of river beds, earthcuts and dikes cannot be truly reproduced.

• It is difficult to extrapolate and interpolate the results obtained from the distorted models.

• The observer may experience an unfavourable psychological effect.

Similitude is defined as the similarity between the model and its prototype in every aspect. Following three types of similarities must exist between the model and prototype.

• Geometric similarity

• Kinematic similarity

• Dynamic similarity

Geometric Similarity:

The model and prototype are said to be geometrically similar if  the corresponding length in the model and prototype are same and the included angles between two corresponding sides are also similar.

Let,

`L_m=` length of model

`H_m=`height of model

`D_m=`diameter of model

`A_m=`Area of model

`V_m=`Volume of model

and `L_p,H_p,D_p,A_p,V_p` are the corresponding values of the prototype.

Now, for geometrical similarity,

`L_m/L_p=B_m/B_p=H_m/H_p=D_m/D_p=L_r`

where, `L_r` is the scale ratio or scale factor.

Similarly, `A_r=A_m/A_p=(L_r)^2`

where `A_r` is the area ratio.

Also, `V_r=V_m/V_p=(L_r)^3`

Where, `V_r` is the volume ratio.

Kinematic Similarity:

The flows in the model and prototype are said to be kinematically similar if at the corresponding points in the model and prototype, the velocity or acceleration ratios are same and the velocity or acceleration vectors points in the same direction. It is the similarity of motion.

`(V_1)_m=`velocity of fluid at point 1 in the model.

`(V_2)_m=`velocity of fluid at point 2 in the model.

`(a_1)_M=`acceleration of fluid at point 1 in the model.

`(a_2)_M=`acceleration of fluid at point 2 in the model.

and `(V_1)_p, (V_2)_p, (a_1)_p, (a_2)_p` are the corresponding values at the corresponding points. 

Now, for Kinematic Similarity,

 `(V_1)_m/(V_1)_p=(V_2)_m/ (V_2)_p=V_r`

`V_r=`Velocity ratio

`(a_1)_M/ (a_1)_p=(a_2)_M/(a_2)_p=a_r`

where, `a_r=`acceleration ratio

• The direction of the velocities in the model and prototype must be same.

• The geometric similarity is a prerequisite for kinematic similarity.

Dynamic Similarity:

The flows in the model and prototype are said to be dynamically similar if at the corresponding points in the model and prototype, the force are same in  ratios and are parallel to each other. It is the similarity of forces.

Let,

`(F_i)_m=`Inertial force at a point in a model.

`(F_v)_m=`Viscous force at a point in the model.

`(F_g)_m=` Gravity force at a point in the model.

and `(F_i)_p`, `(F_v)_p`, `(F_g)_p` be the corresponding values of forces at the corresponding points in prototype.

Now, for dynamic similarity,

`(F_i)_m/(F_i)_p=(F_v)_m/(F_v)_p=(F_g)_m/(F_g)_p=...=F_r=Fo\rce\ ratio`

• The direction of the corresponding forces at the corresponding points in the model and prototype should be same.

• The geometric similarity and kinematic similarity is a prerequisite for dynamic similarity.

Explain the laws of models and write down the criteria of using them.

To ensure the dynamic similarity between model and prototype, it is necessary that the ratio of corresponding forces acting at the corresponding points in the model and prototype be made equal. It implies that dimensionless numbers should be the same for the model as well as prototype . this condition is difficult to be satisfied for all dimensionless numbers. Hence, models are designed on the basis of the forces which is dominating the flow situation.

The laws on which models are designed for dynamic similarity are called model or similarity laws. These are:

• Reynolds Model Law

• Froudes Model Law

• Euler Model Law

• Weber Model Law

• Mach Model Law

Reynolds Model Law:

When the viscous force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Reynolds number is same for both the system. This is known as Reynolds Model Law. i.e,

`(R_e)_(model)=(R_e)_(prot\otype)`

`(rho_mv_ml_m)/mu_m=(rho_pv_pl_p)/mu_p`

`rho_p/rho_m*v_p/v_m*l_p/l_m*mu_m/mu_p=1`

`(rho_r*v_r*l_r)/mu_r=1`

Also,

Time scale ratio, `T_r=l_r/v_r`

Acceleration scale ratio, `a_r=v_r/T_r`

Force scale ratio, `F_r=m_r*a_r=rho_r*A_r*v_r*a_r=rho_r*(l_r)^2*v_r*a_r`

Discharge scale ratio, `Q_r=(rho*A*v)_r=rho_r*(l_r)^2*v_r`

Following are some of the flow phenomenon for which Reynold model law can be sufficient criterion for similarity of flow in the model and prototype:

• Motion of airplanes.

• Flow of incompressible fluid in closed pipes.

• Motion of submarines completely under water.

• Flow around structures or other bodies immersed completely under moving fluids.

Froudes Model Law:

When the gravitational force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Froudes number is same for both the system. This is known as Froudes Model Law. i.e,

`(F_r)_(model)=(F_r)_(prot\otype)`

`v_m/root(2)(l_m*g)=v_p/root(2)(l_p*g)`

`v_p/v_m=root(2)(l_p/l_m)=root(2)(l_r)`

Time scale ratio, `T_r=T_p/T_m=(l_p/v_p)/(l_m/v_m)=l_r*1/root()(l_r)=root()(l_r)`

Acceleration scale ratio, `a_r=a_p/a_m=v_p/T_p*T_m/v_m=root()(l_r)*1/root()(l_r)=1`

Discharge Scale ratio, `Q_r=Q_p/Q_m=(l_p)^3/T_p*T_m/(l_m)^3=l_r^2.5`

Force Scale ratio, `F_r=F_p/F_m=(rho_p*l_p^3*v_p/T_p)/(rho_m*l_m^3*v_m/T_m)=rho_r*l_r^3*1=rho_r*l_r^3`

Pressure Scale ratio, `P_r=P_p/P_m=F_p/F_m*A_m/A_p==rho_r*l_r^3*1/l_r^2=rho_r*l_r`

Energy Scale ratio, `E_r=E_p/E_m=F_p/F_m*L_p/L_m=rho_r*l_r^4`

Momentum or impulse Scale ratio, `M_r=m_p/M_m=(rho_p*l_p^3*v_p)/(rho_m*l_m^3*v_m)=rho_r*l_r^3*root()(l_r)=rho_r*(l_r)^(7/2)`

Torque Scale ratio, `T_r=F_p/F_m*l_p/l_m=l_r^4`

Power Scale Ratio, `p_r=P_p/P_m=F_p/F_m*v_p/v_m=rho_r*l_r^3.5` 

NOTE: `rho_r=1` for same fluid used in both model and prototype.

Following are some of the flow phenomenon for which Froudes model law can be sufficient criterion for similarity of flow in the model and prototype:

• Free surface flows such as flow over spillways, sluices etc.

• Flow of jet from an orifice or nozzle.

• Flow where flow waves are likely to be formed on the surface.

• Flow where fluids of different mass densities flow over one another.

Eulers Model Law:

When the pressure force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Eulers number is same for both the system. This is known as Eulers Model Law. i.e,

`(E_u)_(model)=(E_u)_(prot\otype)`

`v_m/root(2)(p_m/rho_m)=v_p/root(2)(p_p/rho_p)`

For same fluid, `rho_m=rho_p`

`v_m/root(2)(p_m)=v_p/root(2)(p_p)`

This law is applied in the following flow problems:

• When the phenomenon of cavitation occurs.

• Enclosed fluid system where the turbulence is fully developed so that viscous forces are negligible and also force of gravity and surface tension are entirely absent.

Weber Model Law:

When the Surface Tension force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Webers number is same for both the system. This is known as Webers Model Law. i.e,

`(W_e)_(model)=(W_e)_(prot\otype)`

`v_p/root(2)(sigma_p/(rho_p*l_p))=v_m/root(2)(sigma_m/(rho_m*l_m))`

Weber Model Law is applied in following flow situations:

• Flow over weirs involving very low heads.

• Very thin sheet of liquid flowing over a surface.

• Capillary waves in channel.

• Capillary rises in narrow passages.

• Capillary movement of water in soil. 

Mach Model Law:

When the elastic force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Machs number is same for both the system. This is known as Machs Model Law. i.e,

`M_(model)=M_(prot\otype)`

`v_p/root(2)(K_p/rho_p)=v_m/root(2)(K_m/rho_m)`

Mach Model law is applied in following flow situations:

• Aerodynamic testing

• Phenomena involving velocities exceeding the speed of sound.

• Hydraulic model testing for the cases of unsteady flow, especially water hammer problems.

• Under water testing of torpedoes.

Similitude is defined as the similarity between the model and its prototype in every aspect. Following three types of similarities must exist between the model and prototype.

• Geometric similarity

• Kinematic similarity

• Dynamic similarity

Geometric Similarity:

The model and prototype are said to be geometrically similar if  the corresponding length in the model and prototype are same and the included angles between two corresponding sides are also similar.

Let,

`L_m=` length of model

`H_m=`height of model

`D_m=`diameter of model

`A_m=`Area of model

`V_m=`Volume of model

and `L_p,H_p,D_p,A_p,V_p` are the corresponding values of the prototype.

Now, for geometrical similarity,

`L_m/L_p=B_m/B_p=H_m/H_p=D_m/D_p=L_r`

where, `L_r` is the scale ratio or scale factor.

Similarly, `A_r=A_m/A_p=(L_r)^2`

where `A_r` is the area ratio.

Also, `V_r=V_m/V_p=(L_r)^3`

Where, `V_r` is the volume ratio.

Kinematic Similarity:

The flows in the model and prototype are said to be kinematically similar if at the corresponding points in the model and prototype, the velocity or acceleration ratios are same and the velocity or acceleration vectors points in the same direction. It is the similarity of motion.

`(V_1)_m=`velocity of fluid at point 1 in the model.

`(V_2)_m=`velocity of fluid at point 2 in the model.

`(a_1)_M=`acceleration of fluid at point 1 in the model.

`(a_2)_M=`acceleration of fluid at point 2 in the model.

and `(V_1)_p, (V_2)_p, (a_1)_p, (a_2)_p` are the corresponding values at the corresponding points. 

Now, for Kinematic Similarity,

 `(V_1)_m/(V_1)_p=(V_2)_m/ (V_2)_p=V_r`

`V_r=`Velocity ratio

`(a_1)_M/ (a_1)_p=(a_2)_M/(a_2)_p=a_r`

where, `a_r=`acceleration ratio

• The direction of the velocities in the model and prototype must be same.

• The geometric similarity is a prerequisite for kinematic similarity.

Dynamic Similarity:

The flows in the model and prototype are said to be dynamically similar if at the corresponding points in the model and prototype, the force are same in  ratios and are parallel to each other. It is the similarity of forces.

Let,

`(F_i)_m=`Inertial force at a point in a model.

`(F_v)_m=`Viscous force at a point in the model.

`(F_g)_m=` Gravity force at a point in the model.

and `(F_i)_p`, `(F_v)_p`, `(F_g)_p` be the corresponding values of forces at the corresponding points in prototype.

Now, for dynamic similarity,

`(F_i)_m/(F_i)_p=(F_v)_m/(F_v)_p=(F_g)_m/(F_g)_p=...=F_r=Fo\rce\ ratio`

• The direction of the corresponding forces at the corresponding points in the model and prototype should be same.

• The geometric similarity and kinematic similarity is a prerequisite for dynamic similarity.

Explain the laws of models and write down the criteria of using them.

To ensure the dynamic similarity between model and prototype, it is necessary that the ratio of corresponding forces acting at the corresponding points in the model and prototype be made equal. It implies that dimensionless numbers should be the same for the model as well as prototype . this condition is difficult to be satisfied for all dimensionless numbers. Hence, models are designed on the basis of the forces which is dominating the flow situation.

The laws on which models are designed for dynamic similarity are called model or similarity laws. These are:

• Reynolds Model Law

• Froudes Model Law

• Euler Model Law

• Weber Model Law

• Mach Model Law

Reynolds Model Law:

When the viscous force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Reynolds number is same for both the system. This is known as Reynolds Model Law. i.e,

`(R_e)_(model)=(R_e)_(prot\otype)`

`(rho_mv_ml_m)/mu_m=(rho_pv_pl_p)/mu_p`

`rho_p/rho_m*v_p/v_m*l_p/l_m*mu_m/mu_p=1`

`(rho_r*v_r*l_r)/mu_r=1`

Also,

Time scale ratio, `T_r=l_r/v_r`

Acceleration scale ratio, `a_r=v_r/T_r`

Force scale ratio, `F_r=m_r*a_r=rho_r*A_r*v_r*a_r=rho_r*(l_r)^2*v_r*a_r`

Discharge scale ratio, `Q_r=(rho*A*v)_r=rho_r*(l_r)^2*v_r`

Following are some of the flow phenomenon for which Reynold model law can be sufficient criterion for similarity of flow in the model and prototype:

• Motion of airplanes.

• Flow of incompressible fluid in closed pipes.

• Motion of submarines completely under water.

• Flow around structures or other bodies immersed completely under moving fluids.

Froudes Model Law:

When the gravitational force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Froudes number is same for both the system. This is known as Froudes Model Law. i.e,

`(F_r)_(model)=(F_r)_(prot\otype)`

`v_m/root(2)(l_m*g)=v_p/root(2)(l_p*g)`

`v_p/v_m=root(2)(l_p/l_m)=root(2)(l_r)`

Time scale ratio, `T_r=T_p/T_m=(l_p/v_p)/(l_m/v_m)=l_r*1/root()(l_r)=root()(l_r)`

Acceleration scale ratio, `a_r=a_p/a_m=v_p/T_p*T_m/v_m=root()(l_r)*1/root()(l_r)=1`

Discharge Scale ratio, `Q_r=Q_p/Q_m=(l_p)^3/T_p*T_m/(l_m)^3=l_r^2.5`

Force Scale ratio, `F_r=F_p/F_m=(rho_p*l_p^3*v_p/T_p)/(rho_m*l_m^3*v_m/T_m)=rho_r*l_r^3*1=rho_r*l_r^3`

Pressure Scale ratio, `P_r=P_p/P_m=F_p/F_m*A_m/A_p==rho_r*l_r^3*1/l_r^2=rho_r*l_r`

Energy Scale ratio, `E_r=E_p/E_m=F_p/F_m*L_p/L_m=rho_r*l_r^4`

Momentum or impulse Scale ratio, `M_r=m_p/M_m=(rho_p*l_p^3*v_p)/(rho_m*l_m^3*v_m)=rho_r*l_r^3*root()(l_r)=rho_r*(l_r)^(7/2)`

Torque Scale ratio, `T_r=F_p/F_m*l_p/l_m=l_r^4`

Power Scale Ratio, `p_r=P_p/P_m=F_p/F_m*v_p/v_m=rho_r*l_r^3.5` 

NOTE: `rho_r=1` for same fluid used in both model and prototype.

Following are some of the flow phenomenon for which Froudes model law can be sufficient criterion for similarity of flow in the model and prototype:

• Free surface flows such as flow over spillways, sluices etc.

• Flow of jet from an orifice or nozzle.

• Flow where flow waves are likely to be formed on the surface.

• Flow where fluids of different mass densities flow over one another.

Eulers Model Law:

When the pressure force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Eulers number is same for both the system. This is known as Eulers Model Law. i.e,

`(E_u)_(model)=(E_u)_(prot\otype)`

`v_m/root(2)(p_m/rho_m)=v_p/root(2)(p_p/rho_p)`

For same fluid, `rho_m=rho_p`

`v_m/root(2)(p_m)=v_p/root(2)(p_p)`

This law is applied in the following flow problems:

• When the phenomenon of cavitation occurs.

• Enclosed fluid system where the turbulence is fully developed so that viscous forces are negligible and also force of gravity and surface tension are entirely absent.

Weber Model Law:

When the Surface Tension force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Webers number is same for both the system. This is known as Webers Model Law. i.e,

`(W_e)_(model)=(W_e)_(prot\otype)`

`v_p/root(2)(sigma_p/(rho_p*l_p))=v_m/root(2)(sigma_m/(rho_m*l_m))`

Weber Model Law is applied in following flow situations:

• Flow over weirs involving very low heads.

• Very thin sheet of liquid flowing over a surface.

• Capillary waves in channel.

• Capillary rises in narrow passages.

• Capillary movement of water in soil. 

Mach Model Law:

When the elastic force can be considered to be the only predominant force which control the motion in addition to the inertial force, then the similarity of flow in any two such systems can be established if the Machs number is same for both the system. This is known as Machs Model Law. i.e,

`M_(model)=M_(prot\otype)`

`v_p/root(2)(K_p/rho_p)=v_m/root(2)(K_m/rho_m)`

Mach Model law is applied in following flow situations:

• Aerodynamic testing

• Phenomena involving velocities exceeding the speed of sound.

• Hydraulic model testing for the cases of unsteady flow, especially water hammer problems.

• Under water testing of torpedoes.

SOLUTION:

The resistance per unit area depends upon:

• Density `rho`

• Viscosity `mu`

• Velocity `V`

• Diameter `D`

Mathematically,

`F=f(rho,V,D,mu)`

Or, `F=K*rho^a*V^b*D^c*mu^d`

Where `K` is the dimensionless parameter and `a,b,c,d` are the exponents to be determined.

The equation for dimension is,

`ML^-1T^-2=K*[ML^-3]^a*[LT^-1]^b*[L]^c*[ML^-1 T^-1]^d`...(I)

Now, for dimensional homogeneity, the exponents of each dimension on both sides must be equal.

`1=a+d`..(i)

`-1=-3a+b+c-d`...(ii)

`-2=-b-d`...(iii)

Here we have four unknowns but only three equation. therefore, it is not possible to find the values of variables `a,b,c,d`. However, three of them can be expressed in terms of fourth variable.

Thus,

`a=1-d`

`b=2-d`

and `-3+3d+2-d+c-d=-1`

or,`c=-d`

Thus,

`F=K*rho^(1-d)*V^(2-d)*D^-d*mu^d`

`F=K* rho V^2*(mu/(rhoVD))^d`

`F=K*rhoV^2*(R_e)^-d`

`F=rhoV^2*phi(R_e)`

Hence, proved.

SOLUTION:

The resistance per unit area depends upon:

• Density `rho`

• Viscosity `mu`

• Velocity `V`

• Diameter `D`

Mathematically,

`F=f(rho,V,D,mu)`

Or, `F=K*rho^a*V^b*D^c*mu^d`

Where `K` is the dimensionless parameter and `a,b,c,d` are the exponents to be determined.

The equation for dimension is,

`ML^-1T^-2=K*[ML^-3]^a*[LT^-1]^b*[L]^c*[ML^-1 T^-1]^d`...(I)

Now, for dimensional homogeneity, the exponents of each dimension on both sides must be equal.

`1=a+d`..(i)

`-1=-3a+b+c-d`...(ii)

`-2=-b-d`...(iii)

Here we have four unknowns but only three equation. therefore, it is not possible to find the values of variables `a,b,c,d`. However, three of them can be expressed in terms of fourth variable.

Thus,

`a=1-d`

`b=2-d`

and `-3+3d+2-d+c-d=-1`

or,`c=-d`

Thus,

`F=K*rho^(1-d)*V^(2-d)*D^-d*mu^d`

`F=K* rho V^2*(mu/(rhoVD))^d`

`F=K*rhoV^2*(R_e)^-d`

`F=rhoV^2*phi(R_e)`

Hence, proved.