Fluid Mechanics
By definition, a fluid is a material continuum that is unable to withstand a static shear stress. Unlike an elastic solid which responds to a shear stress with a recoverable deformation, a fluid responds with an irrecoverable flow.
Variables needed to define a fluid and its environment are:

Quantity Symbol Object Units
pressure p scalar N/m2
velocity v vector m/s
density r scalar kg/m3
viscosity m scalar kg/m-s
body force b vector N/kg
time t scalar s

Examples of fluids include gases and liquids. Typically, liquids are considered to be incompressible, whereas gases are considered to be compressible. However, there are exceptions in everyday engineering applications.


Types of Flow; Reynolds Number
Fluid flow can be either laminar or turbulent. The factor that determines which type of flow is present is the ratio of inertia forces to viscous forces within the fluid, expressed by the nondimensional Reynolds Number,
where V and D are a fluid characteristic velocity and distance. For example, for fluid flowing in a pipe, V could be the average fluid velocity, and D would be the pipe diameter.
Typically, viscous stresses within a fluid tend to stabilize and organize the flow, whereas excessive fluid inertia tends to disrupt organized flow leading to chaotic turbulent behavior.
Fluid flows are laminar for Reynolds Numbers up to 2000. Beyond a Reynolds Number of 4000, the flow is completely turbulent. Between 2000 and 4000, the flow is in transition between laminar and turbulent, and it is possible to find subregions of both flow types within a given flow field.

Governing Equations
Laminar fluid flow is described by the Navier-Stokes equations. For cases of inviscid flow, the Bernoulli equation can be used to describe the flow. When the flow is zero (i.e. statics), the fluid is governed by the laws of fluid statics.
Open Channel FlowThe Manning Equation is the most commonly used equation to analyze open channel flows. The Manning Equation is utilized in our open channel design calculations - Design of Circular Culverts, Design of Rectangular Channels, and Design of Trapezoidal Channels. It is a semi-empirical equation for simulating water flows in channels and culverts where the water is open to the atmosphere, i.e. not flowing under pressure, and was first presented in 1889 by Robert Manning. The channel can be any shape - circular, rectangular, triangular, etc. The units in the Manning equation appear to be inconsistent; however, the value k has hidden units in it to make the equation consistent. The Manning Equation was developed for uniform steady state flow. Uniform means that the channel is prismatic. Prismatic means the channel has constant dimensions (including depth) along its length. Steady state means the flowrate, velocity, and everything else are constant with time. In reality no flow can be uniform and steady. However, for individual channel reaches (e.g. one mile of a 200 mile river) the assumptions may be fairly well achieved. The Manning Equation is also used successfully to simulate "gradually varied flow" where the channel is not prismatic. In this case, S is the slope of the energy grade line. For prismatic flows, S is the slope of the hydraulic grade line which is the slope of the water surface and is the same as the slope of the channel bottom.
Flows Under Pressure (Closed Conduits, Pipes)
This blog has two comprehensive calculations for simulating steady flows under pressure - Design of Circular Water Pipes and Design of Circular Liquid or Gas Pipes. We also have a water hammer calculation that predicts pressures during transient conditions due to closing or opening a valve. In addition, we have calculations for flow measurement using orifices, nozzles, and venturi meters. The website also has several smaller calculations which solve individual portions of the energy equation; these smaller calculations are linked in the following paragraph.
The energy equation represents elevation, pressure, and velocity forms of energy. The energy equation for a fluid moving in a closed conduit is written between two locations at a distance (length) L apart. Energy losses for flow through ducts and pipes consist of major losses and minor losses. Major losses are due to friction between the moving fluid and the inside walls of the duct. Minor losses are due to fittings such as valves and elbows. Major losses are computed using either the Darcy-Weisbach friction loss equation (which utilizes the Moody friction factor) or the Hazen-Williams friction loss equation. The Darcy-Weisbach method is generally considered more accurate than the Hazen-Williams method. Additionally, the Darcy-Weisbach method is valid for any liquid or gas; Hazen-Williams is only valid for water at ordinary temperatures (40 to 75 oF). The Hazen-Williams method is very popular, especially among civil engineers, since its friction coefficient (C) is not a function of velocity or duct (pipe) diameter. Hazen-Williams is simpler than Darcy-Weisbach for calculations where you are solving for flowrate, velocity, or diameter.