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
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.
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