Methods Of Obtaining Flow Nets

Methods Of Obtaining Flow Nets

The following methods are available for determining flow traps:

1. Graphical solution by sketching
2. Mathematical or analytical methods
3. Numerical analysis
4. Models
5. Analogy Methods
All methods are based on the Laplace equation.

1. Graphical solution by sketching

To obtain the net of a specific cross-section flow, one typically begins by converting the cross-section, particularly when dealing with anisotropic subcells. Then, through a process of trial and error, the boundary conditions are observed.

Flow nets exhibit various properties, such as the arrangement of flow lines and the orthogonality of equipotential lines. It is crucial to adhere to specific rules governing boundary conditions and ensure smooth transitions.

The approach of sketching through trial and error was initially published by Forschheimer in 1930, with further suggestions provided by A. Developer, as proposed by Casagrande in 1937.

Casagrande offered the following recommendations to aid the cartoonist:

(a) Take advantage of every opportunity to study well-constructed flow networks, as it helps in gaining problem-solving experience.

(b) Initially, four to five flow channels are usually sufficient for the first attempt.

(c) It is advisable to sketch the entire flow net roughly before focusing on adjusting the details.

(d) Keep in mind that all transitions should be smooth and possess an oval or parabolic shape.

(e) Begin by identifying and drawing the boundary flow lines and boundary equipotentials.

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This method has the advantage of helping the sketcher get a feel of the problem. An undesirable feature is that the technique is difficult.

Many people are not inherently talented in sketching. This problem is partially offset by the fact that the solution of the two-dimensional flow problem is a relatively insensitive pleasure to the net quality of the flow.

A coarse-flow net also generally allows accurate determination of leakage, pore pressure, and gradient. Additionally, the literature available on geotechnical engineering includes well-flow networks for several general scenarios.

2. Mathematical or analytical methods

In some relatively simple cases the boundary conditions can be expressed by equations, and Laplace’s equation solutions can be obtained mathematically.

The method is of academic interest because of its relatively simple problems and the complexity of mathematics.

Methods Of Obtaining Flow Nets

Perhaps the most famous theoretical solution was given by Kozeni (1933) and then by A. Casagrande expanded further, allowing the ground to flow through the earth dam with a filter drain at the bottom.

However, this flower contains confocal parabolas. Another problem with which the theoretical solution is available in the sheet pile wall (Harr, 1962).

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3. Numerical analysis

When a mathematical solution becomes challenging, numerical techniques can be employed to solve the two-dimensional flow Laplace equation. In particular, relaxation methods are utilized, which involve sequentially estimating the total heads at different points in the mesh or network.

The differential equation of Laplace is transformed into its finite difference form, and the solution is obtained rapidly using a digital computer.

Transition words: When, numerical techniques can be employed, In particular, relaxation methods are utilized, The differential equation is transformed, and the solution is obtained.

4. Models

Researchers can study the flow problem by constructing a scaled model and analyzing the flow within it. For the determination of flow lines, earth dam models are commonly utilized. These models are typically built between two parallel glass or lucite sheets.

By injecting colored spots at various levels, researchers can detect flow lines and directly determine the upper flow curve. Piezometer tubes can also be employed to measure heads at different levels.

However, the use of models is primarily suitable for explaining the fundamental concepts of fluid flow. Building such models poses challenges in terms of time, effort, and capillary difficulties, limiting their applicability in solving general flow problems.

It should be noted that the capillary flow in the zone above the upper flow line may have significance in the model, although it is less prominent in real-world prototypes.

5. Analogy Methods

Laplace’s equation, which applies to fluid flow, also finds application in electric and heat fluxes. The use of power models is increasingly gaining popularity for solving complex fluid flow problems.

In the electrical model, the voltage corresponds to the total head, the current relates to velocity, and conductivity corresponds to permeability.

A similarity can be observed between Ohm’s law and Darcy’s law. By measuring the voltage, one can determine the equipotentials, which in turn allows for drawing the flow pattern.

The power analogy’s versatility makes it a suitable method for resolving complex flow situations. This is particularly advantageous when dealing with boundary conditions that prove to be too challenging for other methods.

Methods Of Obtaining Flow Nets

However, electrical models are considered favorable for instructional purposes, especially with respect to the determination of upper flow lines and flow traps for earth dams.

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