OPEN CHANNEL FLOW

Who Cares!? • flow through creeks, streams, rivers, and drainage channels are all open channel flow examples, without these flows we would not be here OPEN CHANNEL FLOW [ physical interpretation: what are we doing today? ] • Any liquid flowing in a conduit or channel that is not completely filled and sealed (open to atmosphere) is considered an open channel flow • Typically these flows are driven purely by gravity as no pressure gradient can be applied while the flow is totally open to the atmosphere • For a fully developed, steady, open channel flow the pressure distribution is essentially hydrostatic • Steady, fully developed flows develop a balance between the shear force resisting the fluid flow over the channel surface and the fluid weight force 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ physical interpretation: examples of open channel flow ] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ fundamentals ] • The range of values the geometric variables can take in an open channel problem usually prohibit the use of any straightforward analytical approach, therefore scaled physical models are usually developed • The existence of a free-surface in open channel flow means that the flow is free to assume many different forms that are not possible in pressurized pipe flow, this further complicates OC flow analysis • As in closed conduit flows, open channel flows can classified as laminar or turbulent, steady or unsteady, we will only examine steady OC flows • OC flows can be further classified by observing the manner in which the fluid depth, y, varies with distance along the channel, x. 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ uniform, rapidly varying, and gradually varying flows ] • A uniform open channel flow is classified as one where the depth does not vary along the channel (dy/dx=0) • When dy/dx <> 0, the OC flow is called non uniform • Non-uniform flows may have flow depths that vary considerably over a short distance, these are called rapidly varying flows (RVF) (dy/dx~1), if the depth changes slowly with distance this is referred to as a gradually varying flow (GVF) (dy/dx<<1) • The influential forces (pressure, weight, shear, inertia, etc.) have relatively different roles depending on whether a flow is uniform, RVF, or GVF 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ uniform, rapidly varying, and gradually varying flows ] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ laminar, transitional, and turbulent OC flow ] • Whether an OC flow is laminar, transitional, or turbulent depends the Reynolds number Re = rVRh/m, here V is the average velocity of the fluid, Rh is the hydraulic radius • A general range for Reynolds number classification is, for Re<500 OC flows are considered laminar, Re>12,500 are turbulent flows, and otherwise are transitional • Laminar OC hydraulic flows are rare because of water has a small viscosity and characteristic lengths within the flow are typically large 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves ] • The free unrestricted surface of OC flows can deform from their undisturbed relatively flat configuration to form waves • Waves will move across an OC surface at different speeds based on their wave height, wave length, as well as the depth of the channel, and fluid velocity among other things • OC flows can also be characterized by how fast waves move in a channel relative to the mean channel velocity, the dimensionless parameter that describes this is called the Froude number, where Fr=V/(gl)0.5, here l, is a characteristic length of the flow • For Froude number ratios less than one the flow is deemed subcritical(or tranquil), at Fr=1 the flow is referred to as critical, and for Fr>1 the flow is termed supercritical (or rapid) 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • Consider a small elementary wave of height, dy, produced by a suddenly moving (initially stationary) end wall with speed dV • After time t=0, (assuming channel calm at t<0), a stationary observer will witness a single wave moving down the channel with wave speed, c, there will be no fluid motion ahead of the wave, and a fluid velocity dV will exist behind the wave 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • If an observer is moving along the channel with speed c, the flow will appear steady as shown below, to this observer the fluid velocity will be V = -c on the observer’s right, and V = (-c + dV) on the observer’s left • The continuity and momentum equations can be applied to the CV shown above to reveal the relationships between the various parameters in the flow 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • We assume uniform, 1D flow, the continuity equation becomes - [1] • We rearrange this as - [2] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • For small amplitude waves (dy<<y) - [3] • Similar to the continuity equation, we can write the momentum equation - [4] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] - [4] • In [4], the flowrate is written - [5] • Further in [4] the pressure forces (assumed hydrostatic) on the channel cross section at (1) and (2) are - [7] - [6] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] - [4] • Again if we impose the assumption of small amplitude waves (dy)2<<ydy, then the momentum equation [4] reduces further to - [8] - [3] • Combining [8] with [3] we can pick up the wave speed, c as - [9] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] - [9] • From [9] we recognize the speed of a small amplitude solitary wave is proportional to the fluid depth, y, and independent of the wave amplitude, dy, the fluid density is not an important parameter, but gravity is • The insignificance of fluid density is a direct result of the fact that this wave motion is a balance between inertial effects (proportional to r) and weight or hydrostatic pressure effects (proportional to g=rg), a ratio of these effects sees the density, r, drop out from consideration 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • Wave speed may also be calculated using the energy and continuity equations rather than the momentum and continuity equations (as previously derived) • For a simple surface wave shown above, witnessed by an observer moving with a wave speed, c, the flow is steady, and the pressure is of course constant for any point on the free surface, if we apply the BE to this friction-free flow - [10] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] • We differentiate [10] to yield - [11] • We can write an expression of continuity as Vy=constant, (the flow must be constant), if we differentiate this expression of continuity we arrive at - [12] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] - [11] - [12] • Once we combine [11] and [12], we can eliminate dV and dy and invoke the fact that V=c in this situation (as the observer is moving with speed c), we arrive back at - [9] • We must again recall that our results were derived for small amplitude waves (dy/y<<1) and that the flow is one dimensional 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: solitary waves ] - [9] • More detailed considerations have revealed that wave speed for finite-sized solitary waves exceeds that given by [9], a more realistic quantification of wave speed is had from [13] - [13] • [13] indicates that the larger the amplitude, the faster the wave travels 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: continuous waves ] • By not limiting ourselves to solitary wave consideration, much more complex wave patterns can be described by examining continuous waves of sinusoidal shape (as shown above) • Complex surface patterns found in nature (open water fetch driven waves) may be described by combining waves of various wavelengths, l, and amplitudes, dy 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: continuous waves ] • We approach this method utilizing a Fourier series where each term in the series represents a wave of different wavelength and amplitude to represent an arbitrary function or free surface shape 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: continuous waves ] • Wave analysis of this type demonstrates that sinusoidal surface waves of small amplitude vary with both wavelength and fluid depth - [14] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: continuous waves ] [14] - [14] • [14] is graphed on the right, wave speed, c, is plotted against the wave length-to-depth aspect ratio • For conditions (like ocean) where depth is much greater than wavelength, y>>l, the wave speed is independent of y and is given by - [15] 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ surface waves: continuous waves ] • For cases where the fluid layer is shallow, y<<l, the wave speed is also given by [9] - [9] • This result will also be produced by [14] as the hyperbolic tangent term goes to 2py/l as y/l0 • These two limiting cases are apparent from graph of [14] • So we recognize now, that in general, for a given depth, long waves travel the fastest, thus we will only consider this limiting case, c=(gy)1.5 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ froude number effects ] • Consider the solitary wave travelling in the figure to the right • If the fluid layer is not moving, to a stationary observer, the wave moves to the right with a speed c relative to the fluid and observer • If the fluid is flowing to the left with a velocity V<c then the wave will travel with a speed c – V relative to the stationary observer • If the fluid is flowing to the left with a velocity V=c then the wave will remain stationary • If the fluid is flowing to the left with a velocity V>c then the wave will be washed to the left with a speed V – c • These relationships can be expressed in a dimensionless manner through use of the Froude number, Fr = V/(gy)0.5 = V/c (the ratio of the fluid velocity to the wave speed) 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ froude number effects ] • To better illustrate how the Froude number describes OC wave front effects, consider the events that follow a rock being thrown into a moving stream • If the stream is stagnant, the ripples will spread in all directions evenly • For scenarios where the stream is moving slowly i.e. (V<c) or (Fr<1), the waves will move upstream (we say that upstream locations are in hydraulic communication with downstream locations), we call these flows subcritical (viscous effects – not yet discussed – will eventually damp out these upstream bound disturbances) • Conversely, if the stream is moving rapidly, V>c, disturbances on the surface downstream of the observer will be washed further downstream, these conditions where Fr>1 are termed supercritical • For special cases of V=c where Fr=1, the upstream propogating waves remain stationary, this is critical flow 87-351 Fluid Mechanics

OPEN CHANNEL FLOW [ froude number effects ] • There exists a loose analogy between OC flow of a liquid and the compressible flow of a gas, the governing dimensionless parameter in each case is the fluid velocity, V, divided by the surface wave speed for open channel flow and by sound wave speed for compressible flow • Similarities exist in the differences between subcritical (Fr<1) vs supercritical (Fr>1) flows, and subsonic (Ma<1) vs supersonic (Ma>1), where Ma is the Ernst Mach number 87-351 Fluid Mechanics

OPEN CHANNEL FLOW