OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples inlude not only hannels as the word is usually used, but also flow aross flat surfaes suh as parking lots or streets, open oean flow, flow exiting dams, et. Like pipe flow, open hannel flow an be laminar or turbulent, and steady or unsteady; it an also be uniform (onstant depth along the hannel) or non-uniform. In one-dimensional open hannel flow, the distane axis is onventionally labeled x, and the depth y. The Reynolds number is often defined as Re = h rvρ / µ, where r h is the hydrauli radius. With this definition, flow is typially laminar if Re < 500 and turbulent if Re >,500; the wide gap reflets both a more gradual transition in open hannels than in pipes, and also variations among hannels in terms of geometry. Flow in river-sized systems is typially turbulent, and that in thin-layer flow along the ground is often laminar. A key feature of open hannel flow is the presene of waves on the liquid surfae. The veloity of suh waves is of partiular interest. To understand the features that ontrol the veloity of waves, imagine a single wave that is moving right-to-left aross the surfae of a water body at a veloity, as shown in the figure on the left below. Note that the wave is just the disturbane of the water surfae shape; the water underneath the wave is essentially stationary, other than the nearly irular rotation that water near the surfae undergoes as the wave passes. Now imagine that the water to the left (with depth y) is aused to move with veloity to the right. This auses the wave to beome stationary relative to an observer who is outside the system and is not moving. (Note that this same senario ould be established by having the observer move with veloity to the left, without movement of the underlying water.) As the water goes under the wave, it slows down in aord with the ontinuity equation, as follows.
= m ρq ρvyb= ρvyb= ρv y + y b Vy = V y + y () Defining the +x diretion as movement to the right, the water loses x-direted momentum as is passes through a CV defined as the spae between points and on the x axis, the depth of the water olumn, and a width b. Therefore, a fore must be applied in the x diretion. The only fores on the water are the pressure fores, so the momentum equation applied to the CV yields: F = m V p A p A = m V V () y yb γ γ y y y y b ρq V V [ + ] ([ + ] ) = ( ) ( [ ] γ ) γ b y y + y = g V yb V V ( y [ ] y y ) = ( Vy)( V V) g y y + y = V y V V g Substituting the expression from ontinuity from above and arrying out some algebra: Vy y y+ y = ( Vy) V g y + y y y+ y y+ y = Vy Vy V y+ y g y y+ y y+ y = Vy V y g y + y y + y = V y g
y + y g y + y = V y Finally, assuming that y << y and letting V =, we obtain: gy = (3a) = gy (3b) This result is perhaps surprising, in that is indiates that the veloity of the wave relative to the underlying, stationary water depends only on the total depth of the water and is independent of the properties of the fluid (in partiular, ρ and µ) and of the amplitude of the wave (as long as the onstraint that y << y applies),. The derivation is also restrited to shallow water waves, in whih the effets of the wave motion are sensed at the bottom of the hannel. For deep water waves (e.g., in the oean), the veloity beomes muh less than gy and is given by the following figure (Fig. 0.5 from Munson): If we now imagine the same senario as above but allow the water far to the left to move at a veloity anywhere from zero to values greater than, we see that the net movement of the wave ould be either to the left (for V < ) or to the right (for V > ). In the latter ase, the existene of the wave (or any suh disturbane) is never sensed by the fluid to the left. The distintion between the two situations in whih disturbanes an and annot propagate upstream is an important one, so the two onditions are given distintive names: when the fluid veloity is greater than the wave veloity, the onditions are said to be superritial, and when the fluid veloity is less than the wave veloity, the onditions are subritial. A key dimensionless number for open hannel flow is the Froude number, whih an be interpreted in general as the ratio of the inertial fore on the water to the gravitational fore. In 3
general, Fr V / ( gl) / =, where l is a harateristi length. In open hannel flow, l is taken as the depth of flow, so the Froude number expresses the ratio of the flow veloity to the veloity of a surfae wave. In the speial ase where Fr =, surfae waves remain stationary and the onditions are said to be ritial. The Froude number is equally or more important than Re for open hannel flow. To explore the behavior of flow in open hannels quantitatively, we begin with (and often model more omplex systems as) onsideration of in one-dimensional flow through a simple retangular ross-setion of width b. We designate the flow rate per unit hannel width (Q/b) as q, and the elevation of the bottom of the hannel relative to a speified datum as z bot. We assume that b is onstant, so q is onstant as well. However, the hannel bottom might or might not be horizontal; therefore, for the general ase z bot is a funtion of x. The veloity an be related to q and b by: V x Q qb q = = = (4) A x y x b y x Beause we are assuming that the fluid is ideal, the total energy per unit weight (i.e., the total head) must remain onstant both with depth and along the flow path. Furthermore, if we onsider a streamline along the water surfae, the pressure head is zero everywhere. Therefore the total head at the water surfae is just the sum of the elevation head (h elev ) and the veloity head (h vel ). The total head at any depth is the same as the total head at the surfae, so we an write: h = h = h + h + h vel, surf (5) tot tot, surf elev, surf pressure, surf V x q htot = zbot ( x) + y( x) + = zbot ( x) + y( x) + g g y( x) (6) Or, equivalently, in terms of the onditions at the bottom of the water olumn (assuming that the pressure distribution is hydrostati, equivalent to assuming that the flow is uniform): pbot x q htot = zbot ( x) + + γ g y( x) (7) Note that, even though the terms on the right in the two preeding equations all vary with x, h tot is independent of x. In the remainder of the disussion, we will drop the expliit indiation that z bot and y are funtions of x. The sum of the veloity head and the depth (or the veloity head and the pressure head at the bottom of the hannel) at a given loation is referred to as the speifi energy, E. Thus, the speifi energy at any loation where the flow is uniform an be expressed as: 4
q pbot q E = y+ = + = h tot zbot (8) gy γ gy As indiated by the final equality in Equation 8, for the given assumptions (fixed Q and b and steady, ideal flow), the speifi energy is idential at all loations where z bot is the same, but it differs at loations where the elevation of the hannel bottom differs. Thus, the speifi energy is the variable portion of the head at a given loation; the first equality in Equation 8 indiates that this value depends only on the depth of the water at that loation. For a given q in a given hannel, the flow ould have any depth whatsoever; the only onstraint is that Equation 4 must be satisfied. Aording to Equation 8, for eah depth, the water has a unique value of speifi energy. A plot of E vs. y for a given q has a harateristi shape, approahing the asymptotes of y = E and y = 0 as E gets large, and passing through a minimum value of E at some intermediate value of y. Conventionally, suh plots are drawn with y as the vertial axis (to orrespond to our intuitive way of thinking about depth), even though y is probably a more logial independent parameter. The plot is shown in both ways in Figure. Figure. A typial speifi energy diagram. The analogous parameter for onfined flow with a fixed Q and fixed system geometry would be the pressure head, sine that is the only type of head that an hange at a given loation in suh a system. Beause of this, in onfined systems, there is no point in defining a separate term analogous to speifi energy. 5
The value of y at whih the speifi energy passes through its minimum an be found by differentiating Equation 8 with respet to y and setting the result to zero. Designating this value of y as y, we find: de dy q = = 0 3 gy 3 q = gy y q = g /3 (9) The orresponding values of E min and the fluid veloity are: q q 3 E y y y y y V 3 min = + = + = + = gy g y y 3/ / ( y g ) = q gy y = y = (0) Comparison of the final expression in Equation 0 with that in Equation 3b indiates that the veloity orresponding to the minimum speifi energy is the veloity of a surfae wave in the system. As a result, on the portion of the speifi energy urve where V > V, the flow is superritial, and on the portion where V < V, flow is subritial. To summarize, to sustain any speified flow rate per unit width (Q/A, or q) of an ideal fluid in an open hannel, the speifi energy of the fluid must exeed a minimum value, E min. If E ever dereases below this value, the value of q passing the loation will be less than the value of q approahing it, and the fluid will bak up. This transient event inreases both y and E just upstream, a proess that ontinues until the fluid has attained depth y and speifi energy E min, at whih point steady flow is again ahieved. When the depth is exatly y and the speifi energy is E min, the veloity is gy, orresponding to a Froude number of. If the fluid has any speifi energy that is greater than E min, then uniform flow an be sustained at two different veloities: one subritial and one superritial. In a general sense, when the veloity is subritial, the elevation head is more signifiant than the veloity head, and when the flow is superritial, the reverse is true, although the transition does not orrespond exatly to the point where the two heads are equal. Now onsider an open hannel with a fixed width but a region in whih the bottom has a slope between two regions where it is horizontal. Initially, we will onsider four different senarios: either downward and upward sloping bottom, and with the upstream flow being either sub- or super-ritial. Then we will onsider a fifth senario in whih the water enounters a bump that onsists of an upward slope followed by an equivalent downward one. In all ases, the 6
ombination of the ontinuity and energy equations relating upstream and downstream onditions an be written as: V V zbot, + y + = zbot, + y + g g E = E + z z = E + z () bot, bot, bot Senario : Upward slope, sub-ritial upstream flow. In this senario, the upstream ondition is haraterized by a point on the large y portion of the speifi energy diagram whih asymptotially approahes the line y = E. The downstream loation has less speifi energy, so the shift is away from the y = E line, in the diretion of dereasing depth and therefore inreasing veloity. More speifially, at any value of z bot, the system is haraterized by the point on the sub-ritial part of the E urve where E = E z bot. Beause dy/de is > on this leg of the urve, and the magnitude of the hange in E equals the magnitude of the hange in z bot, the derease in y is greater than the inrease in z bot. Thus, perhaps ounter-intuitively, as the bottom of the hannel gets higher, the surfae of the water gets lower. This proess ontinues until y delines to y. If the bottom ontinues to slope upward (i.e., if the hannel bottom has not beome horizontal by the point where y = y ), it still must be the ase that E = E z bot. However, E annot deline any more at the given value of q, beause it has reahed E min. Therefore, the assumption that was used to develop the y vs. E urve (onstant q) is transiently violated, and q passing the ritial point beomes less than q approahing it. This auses water to bak up upstream, inreasing y and E, until the steady flow of q is re-established. This proess ontinues as long as the hannel bottom ontinues to rise, so that y and E always reah y and E min, respetively, just at the point where the bottom flattens out; from there downstream, the flow remains ritial as long as z bot remains onstant. In essene, the system satisfies Equation by allowing E to derease until it reahes E min, and satisfies the equation thereafter by ausing E to inrease. Senario : Upward slope, super-ritial upstream flow. In this senario, the upstream ondition is haraterized by a point on the low y leg of the speifi energy diagram, along whih the urve approahes y = 0 asymptotially. As in Senario, the downstream loation has less speifi energy than the upstream loation (by z bot ), so the shift is away from the y = 0 asymptote, in the diretion of dereasing veloity and inreasing depth. Thus, in this ase, the water surfae gains elevation as the hannel bottom gets higher, opposite from the ase in Senario. One again, if the bottom elevation inreases enough, the system eventually reahes the limiting ondition of E = E min, and the water baks up. As in Senario, bakup ontinues until ritial onditions are reahed right at the loation where the bottom flattens out. Senarios 3 and 4: Downward slope, sub- or super-ritial upstream flow. These senarios are easier to analyze than the senarios desribed above, beause with a downward sloping bottom, the shift is toward inreasing E, away from E min. As a result, the fluid remains on whihever leg of the speifi energy diagram haraterized the upstream flow, just moving to the right. In the ase of super-ritial flow, the veloity inreases in the downstream diretion, and the depth dereases; sine both the hannel bottom and the depth of water derease, the water surfae goes downhill. By ontrast, and again perhaps ounter-intuitively, if the upstream flow is sub-ritial, the depth inreases more than the bottom elevation drops, and the water surfae inreases; i.e., 7
the water flows uphill! This ours beause the gain of potential energy is ompensated by the derease in kineti energy. Senario 5: Bump in the hannel bottom, modeled as a setion with an upward slope, a flat setion, and then a setion with a downward slope, bak to the original elevation. If a subritial flow approahes a bump, its veloity inreases and its depth dereases as it limbs the uphill slope. If the flow never beomes ritial, then the exat reverse proess ours on the other side of the bump, and the downstream flow harateristis are idential to those of the upstream flow. However, if the ritial ondition is reahed, then the flow an return to E via either the sub-ritial or super-ritial leg of the speifi energy urve. Empirially, whenever E inreases from a ritial flow ondition, the super-ritial path is taken preferentially; some obstrution or other impediment to flow is required for the flow to follow the sub-ritial path. Flow onstritions. The preeding disussion fouses on hanges in the elevation of the hannel bottom (z bot ) in systems with onstant width (b). We next onsider the inverse ase of hanges in b in systems with onstant z bot, orresponding to onstritions or expansions in the flow path. For this analysis, we fous on a onstrition in whih b > b. Assuming, as before, that Q is onstant, we onlude that in this ase q < q. However, if the flow is ideal, E = E (beause Equation applies with z bot = 0). Therefore, the situation is desribed not by one, but two y vs. E urves, one for eah q. Sine the value of q in the onstrition is larger than that upstream, the E urve is to the right of the E urve. As the flow enters the onstrition, the fat that E remains onstant means that the system must move vertially from the q urve to the q urve. Given the relative loations of these urves, it is lear that the shift is to shallower flow if the upstream flow is subritial and to deeper flow if the upstream flow is super-ritial. As the onstrition gets tighter, b dereases and q ontinues to inrease, so the y vs. E urve for the onstrition moves farther to the right. Eventually, it moves so far that the flow in the onstrition beomes ritial. Analogous to the senarios analyzed above, if the onstrition is made even tighter, it annot sustain the flow Q, and the water baks up upstream. As a result, E inreases, until it equals E min in the onstrition, and steady flow is one again established. The same situation an be haraterized on a single graph by plotting y vs. q for the given E, as opposed to y vs. E for a given q. The plot an be prepared by solving Equation for q, whih yields: 3 q = g y E y A typial plot of this equation is shown in Figure. In this ase, the onditions haraterizing flow along the onstriting path orrespond to those that move along the urve from left to right, along the upper leg if the upstream flow is sub-ritial and the lower leg if the flow is superritial. 8
Figure. Plots the depth of flow as a funtion of (a) speifi energy and (b) flow rate per unit width, in systems with open hannel flow. Hydrauli Jumps. If flow in an open hannel is suffiiently rapid and the hannel disharges into a zone of lower veloity, a hydrauli jump ours, in whih the elevation of the water surfae undergoes a dramati inrease over a short distane, aompanied by a great deal of turbulene and air entrainment. Despite the inherent non-ideality of the situation, analyzing a CV that inludes the jump with the impulse-momentum equation, assuming ideal fluid behavior, provides some useful information. A definition sketh of the system is shown below. 9
Applying the impulse-momentum equation between points and, the only external fores ating on the fluid in the CV are the pressure-based fores on its ends, so: γ yb γ yb Fext, x = F F = = Qρ ( V V) () Substituting Q/yb for the V terms and γ /g for ρ, and rearranging: γ yb γ yb γ Q Q = Q g y b yb y y Q Q q q = = g yb yb g y y (3) (4) q gy y q y + = + (5) gy Alternatively, by fatoring a ommon fator y y from both sides of Equation 4, the equation an be written as a quadrati in y /y and solved via the quadrati equation to yield: y 8q 8V = + + 3 = + + y gy gy (6) The solution to Equation 6 is suh that y / y is less than, equal to, or greater than whenever V / gy falls into the same range. A situation in whih y / y is less than is impossible, sine that would orrespond to a spontaneous inrease in the EL (i.e., a negative head loss aross the jump). The situation where y / y equals orresponds to stable flow and 0
no jump. Thus, the only region of interest is where both y / y and V / gy are greater than ; put another way, V / gy must be > for a hydrauli jump to be possible. Beause V / gy is the Froude number (Fr), we onlude that a hydrauli jump is possible only if the Froude number is > (i.e., if the flow is super-ritial).