Model Analysis – Model vs Prototype

The Model Analysis is carried out to analyze or find out the performance of hydraulic structures or hydraulic machines before its construction or manufacturing. Several tests and analysis are made using these models to obtain the required information.

The Model is a scaled replica of a structure or machine and Prototype is the name called to the original structure or machine. The model analysis is very helpful in finding out or solving various flow patterns and complex flow problems, to obtain the possible performance of the prototype, and to solve various design problems.

Classification of Analysis Models in Model Analysis

Hydraulic models are broadly classified into two types:

Undistorted Models

Undistorted models are those which are geometrically similar (scale ratios for linear dimensions of prototype and model are the same). The prototype performance can easily understand or predict by studying undistorted models.

Distorted Model

The models which are not geometrically similar to the prototype are called Distorted Models. In Distorted models, different scale ratios are adopted for linear dimensions. I.e. In the case of hydraulic structures, if the scale ratios of horizontal dimensions and vertical dimensions are not equal for the preferred model, that model is a distorted model.

Commonly adopted scale ratios are given below:

1. Dams and Spillways: 1/30 to 1/400
2. Headworks,  gates, canals: 1/5 to 1/25
3. Rivers, Harbors: 1/100 to 1/1000

Type of Similarities between model and prototype

Similarity has a significant role in model analysis especially in predicting the performance of the prototype i.e. while analyzing a property of a particular component or point in the model, we are able to determine the possible property at the same component or point in the prototype. 

Geometric similarity usually maintained in most of the models for better analysis. But in the case of rivers, dams, harbors the geometric similarity is not feasible. This is because the depth of water and flow rates makes it complex in analysis. The similarity between model and prototype is called similitude and the models should hold up the similarity in every aspect compared to the prototype.

The following three types of similarity should exist between model and prototype:

Geometric similarity 

If the ratio of corresponding linear dimensions of model and prototype is the same then it is called geometric similarity.

L_p/l_m=B_p/b_m=D_p/d_m= L_r   (L_p B_p D_p are linear dimensions of prototype.  l_m b_m d_m are the linear dimensions of model) L_r is the scale ratio.

Kinematic similarity

Kinematic similarity is based on the similarity of motion between model and prototype. If the ratios of velocity and acceleration at a particular point of the model and the corresponding point in the prototype are the same, the kinematic similarity exists there.

For velocity,  V_P1/V_m1=V_P2/V_m2=V_r

For acceleration,  a_p1/a_m1=a_p2/a_m2=a_r

Dynamic similarity

The dynamic similarity is defined as the similarity of forces between model and prototype. If the scale ratio of corresponding forces (also the direction of forces) acting on the corresponding points of model and prototype are the same, then we can assume the existence of dynamic similarity.

(F_i)_P/(F_i)_m=(F_v)_P/(F_v)_m=(F_g)_P/(F_g)_m=F_r  (force ratio)

(F_i)_P = Inertia of force in prototype

(F_v)_P= Viscous force in prototype

(F_g)_P= Gravity force in prototype

(F_i)_m (F_v)_m (F_g)_m are the corresponding forces in model

Applications of Model Analysis & Studies

Dams & Model Analysis

Model tests or studies are a crucial part of dam design and construction. The possible working and performances of spillways, penstocks pipes, gates, etc. and forces acting on it can be studied by model analysis. The model analysis also used to determine the type of dam for a proposed site and possible outcome according to the hydraulic forces, flow rates, and discharge of water.

Rivers and Harbors

In hydraulic engineering works in rivers such) as the widening of the river, dredging, canal constructions, weir construction, etc. model studies are carried out to calculate the possible outcome of these constructions. Discharge control and improvement, seepage calculation, erosion, etc. are analyzed by model studies.

In Harbor construction model analysis is done to study on different tidal cycles and wave action that likely to act on harbors and for design purposes. The model studies also useful in determining various protection systems in harbors from wave actions and possible damages from tides cycles.

Hydraulic machines

Possible performance of hydraulic machines like turbines, pumps, etc. can easily be analyzed by model studies. Model studies are very useful for the design purposes of hydraulic machines.

Hydraulic structures

Various analyses can be carried out using model analysis of hydraulic structures during its design period. Possible deflection, the stability of structure on various force actions, destructive and non-destructive tests, etc. are some of the common analyses done on models of hydraulic structures.

Seepage analysis & Hydraulic Model – Model Analysis

Another important application of model analysis is the studies of seepage flow. Studies on seepage flow are helpful in calculating uplift pressure in hydraulic structures. This is applicable in geometrically similar models.

Model laws

The model laws are based on the dynamic similarity between the model and prototype. To maintain dynamic similarity, the model and prototype should maintain the same ratios of corresponding forces acting on the respective points preferred.

These ratios are dimensionless numbers and hence it is quite difficult to maintain the dimensionless same for both model and prototype. So model laws are designed to maintaining dynamic similarity.

• Reynold’s Model law: Reynolds’s model law is based on Reynold’s number. Reynold’s model number is the ratio of inertia force and viscous force. Hence, if the viscous force is predominant, a model is dynamically similar when Reynold’s number of the model must be equal to Reynold’s number of the prototype.
• Froud’s Model law:  Froud’s model law is based on Froud’s number (defined as the square root of inertia force of flowing fluid to the gravity force) which is applicable when gravity force is predominant. According to Froud’s model law, a model is dynamically similar when the Froud’s number is model should equal to Froud’s number of the prototype when the gravitational force is predominant. Froud’s model law is used to solve complex problems in flows over spillways, channels, weirs, sluice, etc. and also flow jets and wave actions.    
• Euler’s Model law: Euler’s model law is based on Euler’s number (defined as the square root of the ratio of inertia of flowing fluid to pressure force by the fluid). Euler’s law is applicable when pressure force is predominant in addition to the inertia force. I.e. the dynamic similarity can achieve (when pressure force is predominant) if Euler’s number for model and prototype are the same. Euler’s law is used in the design of closed pipe flows and where cavitation takes place.
• Weber Model law: This law based on Weber number (square root of the ratio of inertia of flowing fluid to the tension force), so that if surface tension force is predominant, the dynamic similarity can be achieved when weber number of model and prototype are same. This law is applied in the cases of capillary rise in narrow paths, capillary movement of water, and flow over weirs.
• Mach Model law: Mach law is based on the Match number (square root of the ratio of inertia of force to the elastic force of fluid). According to Mach model law, if force due to elastic compression predominates the dynamic similarity can be achieved when the Mach number of the model and the prototype are the same.

Conclusion on Model Analysis

The above mentioned are some advantages and examples of model analysis. Please note that the model analysis is not the final method for solving complex hydraulic design problems. The model analysis is used to find out or diagnose the possible problems that can occur and its nature.

The results from the model analysis not always 100% reliable and authentic. Combined experimental studies and model analysis should be carried out to conclude the results on hydraulic structural design and construction