encyclopedia of inland waters || pressure

PressureJ F Atkinson University of Buffalo Buffalo NY USA

atilde 2009 Elsevier Inc All rights reserved

Introduction

Pressure is an example of a surface stress (forceper unit area) and is important in many problemsof fluid mechanics as well as in a variety of biologicalresponses in natural water bodies Pressure has animpact on each of the physical chemical and bio-logical components of a fluid system In this articlewe dealmostlywith the analysis of the physical impactsof pressure and provide several examples of biologicalresponses to variations in pressure including algaefish and humansThe study of fluid mechanics is ultimately the study

of the response of a fluid to applied forces subject tocertain constraints such as continuity or mass bal-ance and constitutive functions such as the relation-ship between shear stress and rate of strain Fluidforces can generally be divided into body forces andsurface forces Body forces act on all the fluid withina defined control volume or fluid element whilesurface forces act on the surface that encloses thatvolume Furthermore the surface forces can be sepa-rated into those that act either normal to or in theplane of the surface of interest (Figure 1) Here weconsider both forces and stresses There are manysituations in which analysis on the basis of forces ismore useful For example calculation of forces actingon a pipe bend or forces acting on a sluice gate orcanal lock may be needed in the design of a pipedistribution system or for structures in an open chan-nel flow respectively For differential analysis of fluidflow such as would be used in describing details ofthe velocity distribution stresses are generally ofmore direct interest than forces Stresses acting inthe plane of a surface are called shear stresses andstresses acting perpendicular to a surface are callednormal stresses Normal stress is generally calledpressure and pressure at a point has the same magni-tude in all directions Although by mathematical con-vention the normal stress in Figure 1 is consideredpositive as shown pointing outwards from the fluidelement pressure is considered positive as a compres-sive stress since fluids cannot withstand tensionforces (this is a fundamental difference between themechanical description of fluids and solids) Thuspressure acting on a fluid element as a result of thesurrounding fluid is the negative of the normal stressand points inward on all faces of the elementIn general pressure can be divided into two parts

the hydrostatic and dynamic components Hydrostatic

pressure is generated by the force of gravity acting onfluid that lies above a particular point of interest atwhich the value of pressure is desired and dynamicpressure is due to the movement of a fluid As shownbelow hydrostatic pressure also can be assumed inmany analyses of fluid flow especially in environmen-tal applications For so-calledNewtonian fluids whichencompass most fluids of practical interest includingwater it is assumed that the shear stress is proportionalto the rate of strain which in turn can be approxi-mated by the velocity gradient in the direction normalto the surface on which the shear stress acts Thusin a static fluid or in a fluid that is moving but inwhich there are no velocity gradients (ie no relativemotion) the only forces to consider are the body andnormal forces

In the present article although many of the con-cepts apply to fluids in general it will be assumed thatthe main fluid of interest is water either fresh orsaline The main property of interest that pertains towater and is useful for the analysis of pressure dis-tributions is that of incompressibility meaning thatthe density of the fluid is not a function of pressure Inaddition a normal right-handed Cartesian coordinatesystem will be assumed

Hydrostatic Pressure

As the name implies hydrostatic pressure (heredenoted by ph) is the pressure that is exerted underconditions of no fluid motion To understand how phvaries first consider a small element of fluid within alarger volume of that same fluid (Figure 2) The top ofthis element is at a depth h below the water surfaceSimilar to Figure 1 the element has sides dx dy anddz aligned with each of the respective coordinatedirections The forces acting on the element includeits own weight and the forces of the surrounding fluidacting on the surface of the element Since there isno fluid motion or more precisely no relative motionof the fluid element with respect to the surround-ing fluid the only surface force to consider is thatof pressure If the element is to be in equilibriumand therefore to remain motionless (as a result ofNewtonrsquos First Law) the forces acting on it in eachdirection must be in balance

In Figure 2 the sides of the fluid element are num-bered for convenience with sides 1 and 2 referringto faces with perpendiculars (to the faces) along the

155

156 Properties of Water _ Pressure

x dire ction side s 3 and 4 refe rring to side s withperpendi culars along the y direction and sides 5 and6 with perpendicu lars a long the z dire ction Note thatthe z dire ction is orien ted paral lel to the dire ction ofgravity but points positive upwar ds If ph is con sid-ered to be the press ure at the cente r of the elementthen press ures along each of the faces can beexpres sed in terms of ph using a trunca ted TaylorSeries exp ansion an d force (product of stress a ndarea on which it acts) balanc es in each of the threecoordi nate directio ns may be written as

ethx THORN ph1 ethd y dz THORN frac14 ph2 ethd y d z THORN ) ph ph x

d x

2

frac14 ph thorn ph x

d x

2

frac12 1

eth y THORN ph3 ethd x dz THORN frac14 ph4 ethd x d z THORN ) ph ph y

d y

2

frac14 ph thorn ph y

d y

2

frac12 2

ethz THORN ph5 ethd x d y THORN frac14 ph6 ethd x dy THORN thorn g eth dx d y d z THORN ) ph ph z

d z

2

frac14 ph thorn ph z

d z

2

thorn g d z frac12 3

dx

dy

Normal stress(pressure)

Shear stresses(in plane of surface)

dz

Figure 1 Stresses acting on one surface of a small fluid

element

3

1y

x

zh

We

Figure 2 Forces acting on a small fluid element under static conditioby the rectangular surface and the pressure acting on each of the fac

where ph i is the hyd rostatic press ure actin g on face i gis gravitational acceleration and r is the fluid densityNote that dx dy and dz are assumed to be smallenough that any variations in phi over the area ofthe side may be neglecte d Equa tion [3] differs fromeqns [1] and [2] only in that the fluid weight isinclud ed in the force balanc e Simp lifying eqns[1]ndash[3 ] resul ts in

phx

frac14 phy

frac14 0phz

frac14 g frac14 frac124

where g frac14 rg is the specific weight of the fluidKeeping the assumption of constant density and

letting the fluid be water the pressure on the top ofthe fluid element in Figure 2 is given by integratingthe last part of eqn [4] resulting in p frac14 p0thorn ghwhere p0 is the pressure at the water surface andhfrac14 z0z is the water depth where z0 is the elevationat the water surface and z is the elevation at the top ofthe element Also note that the weight (W) of thewater column above the fluid element is Wfrac14 ghAwhere Afrac14 dxdy is the area of the element If p0frac14 0is assumed then the pressure acting on the topsurface of the fluid element in Figure 2 is simply theweight of water sitting above the element divided bythe area (dxdy) Using similar reasoning the pressureat the water surface is actually given by the weightof air above the surface per unit area (note thatpressure can be defined in terms of weight whetheror not the density is constant but if density is notconstant the weight must be determined by integra-tion of the density distribution) In fact Newton longago deduced that air has mass by determining thatwater could be pumped up a tube by extracting airat the top only to a maximum height of about 33 ft(10m) since the weight of a column of water33 ft high weighs approximately the same as a similarcolumn of air stretching through the entire

5

6

4

2 Pressure

Water surface

ight

ns (no fluid motion) forces include the weight of the fluid enclosedes (six totally) of the element

Properties of Water _ Pressure 157

atmosphere Atmospheric pressure changes slightlywith weather and with location (elevation) but isusually assumed to be 147 psi 1 atm or about101 kPa For convenience the pressure at the watersurface is often taken as zero which facilitates theinterpretation of pressure in terms of weight of waterabove a given point of interest Pressures written withrespect to real air pressure as a reference or boundarycondition are said to be lsquoabsolutersquo pressures whilepressures written in terms of zero pressure at thewater surface are said to be lsquogaugersquo pressures Inabsolute terms pressure at a depth of 33 ft (10m) isapproximately double what it is at the water surfaceand pressure increases by 1 atm for every increase indepth of 33 ft The lowest possible pressure in abso-lute terms is zero which is the pressure in deep spacewhile gauge pressures may take negative values (indi-cating a vacuum) In the remainder of this article allpressures will be assumed to be in lsquogaugersquo termsThe relationship between pressure and depth is

further illustrated by considering the case where noexternal forces are applied (except gravity) and inaddition it is assumed that density and thereforespecific weight is constant as is usually the casewith wat er Then a simple integrat ion of eqn [4]gives the general expression for pressure differencebetween two points at different vertical locations inthe fluid

ph frac14 ph2 ph1 frac14 z frac14 ethz1 z2THORN frac125where z1 and z2 are the two locations at which ph isevaluated (Figure 3)Equa tion [4] describes the dist ribution of press ure

in a static fluid In words pressure in a static fluid isconstant on a horizontal plane and has a gradientin the vertical direction equal to g In other wordspressure decreases while moving upward in a staticfluid at a rate given by g Finding the actual

ph1

ph2Δph

z1

z2

z

Pressure distribution

Δz

Figure 3 Pressure variations in a static fluid with constantdensity

pressure at any point requires integrating the lastpart of eqn [4] while impo sing a bounda ry con ditiongiven by a known pressure at some location aswas done above The direction of the vertical gradientimplies that pressure increases while moving down-ward in the fluid However pressure may be forcedto increase upwards if a fluid were to be subjectedto a downward acceleration with a magnitude greaterthan g For example consider a fluid in a container(so there is no relative movement of fluid particleswith respect to each other ie no velocity gradients)subjected to a downward acceleration of magnitudea while still assuming gravity is effective in whichcase an extra force would have to be includedin eqn [3] The last part of eqn [4] would thenbecome

phz

frac14 ethg THORN frac126

and if a gt g the pressure would increase upwardsin the fluid Similarly if the acceleration wasupward a and g would be additive and the mag-nitude of the pressure gradient would be greaterthan g For a fluid mass in free-fall there is noeffective weight since there is no resistance to grav-ity and eqn [3] woul d result in a zero vert icalgradi ent and press ure would be consta nt every-where within the falling mass (equal to zero ingauge pressure terms)

For fluids being accelerated in the x or y directionsthe appropriate force or stress would have to beinclud ed in either eqn [1] or [2] and in general thecorresponding pressure gradient would no longer bezero If a force were applied to accelerate a fluid in acontainer in a horizontal direction say in the x direc-tion with magnitude ax then eqn [1] woul d have toinclude this force and the resulting gradient would be

phx

frac14 x frac127

If the container had a surface open to the atmospherethe gradi ent in eq n [7] woul d be exh ibited by agradient in surface elevation (Figure 4) where pres-sure along a horizontal line would be directly propor-tional to water depth since there is still no relativemotion of fluid in any part of the container relative tofluid in any other part of the container An example ofthis type of problem would be a tanker truck carryingwater with a free surface and accelerating (or decel-erating) along a highway Using the above results itis possible to calculate the maximum accelerationpossible before water would spill over the sides ofthe tank

The situation is similar for a fluid in solid-bodyrotation Considering a fluid rotating around a verti-cal axis applying a similar force balance in the radial


direction as was previously done in vertical and hori-zontal direction s (eqns [1]ndash[3]) results in

phr

frac14 r2 frac128

where r is the radial coordinate and o is the angularvelocit y Upon integrat ing eqn [8] assuming con stantdensity pressure is found to vary parabolically withradial distance and then considering a surface opento the atmosphere the fluid surface will take a para-bolic shape as well since in this case rotation (about avertical axis) does not affect forces or pressure distri-bution in the vertical direction Although of interestin many applications of fluid mechanics situationsin which a fluid is artificially accelerated either line-arly or in rotation are rare for environmental appli-cations and these situations will not be consideredfurther here

Density Variations

When density is not constant its variation with depthmust be known in order to integrate the last partof eqn [4] to obta in the press ure distri bution Forexample if density increases linearly with depthdrdz frac14 k where k is a (positive) constant then thepressure variation would be quadratic with depth and

ph2 ph1 frac14 gk

2ethz21 z22THORN thorn 0ethz1 z2THORN

frac129

where r0 is the density at zfrac14 0 As previously notedwhen working with water the incompressibility

Increasing denspressure

Fluid element

Figure 5 Fluid element in fluid where lines of constant density are

Direction ofacceleration

hDirection of

pressure gradient

Figure 4 Response of water surface in a container of fluid ofconstant density and with a surface open to the atmosphere and

being accelerated to the right along the bottom p frac14 gh where

h is the depth at any location

assumption may be applied so the density is notaffected by pressure (density of water becomes a func-tion of pressure only in the deepest parts of oceans) andfor inland waters it may be assumed that densitydepends on temperature and salinity only This relation-ship is expressed through an equation of state

frac14 T Seth THORN frac1210whereT is temperature and S is salinity For most inlandwaters salinity is small enough (or zero) that only tem-perature variations are important For fresh water thetemperature of maximum density is 4 C and the varia-tion of density with temperature may be approximatedby a parabolic function such as

frac14 0 1 000663 T 4eth THORN2 frac1211where r0frac14 99997kgm3 is the density at 4 C andT isin (C)

Independe nt of any densi ty stra tificatio n eqn [4]must still hold For example it can easily be shownthat lines of constant density must be horizontal in astatic fluid since equilibrium would not otherwise bepossible To see this consider a force balance appliedto the fluid element as shown in Figure 5 With linesof constant density oriented as shown r increaseswhile movi ng along a horiz ontal line tow ards theright Accordi ng to the last part of eqn [4] the pr es-sure would then also increase while moving along ahorizontal line to the right (assuming a horizontalwater surface) resulting in a non-zero pressure gradi-ent viola ting the first part of eqn [4] This situatio n isimpossible in a static fluid although it may be possi-ble in a fluid moving in a body of water large enoughthat the earthrsquos rotation (ie Coriolis acceleration)could generate a pressure gradient (tilt in watersurface) to counter-balance the pressure gradientassociated with the horizontal density gradient

Hydrostatic Forces on Submerged Surfaces

In addition to development of the fluid equations ofmotion understanding of pressure has practical appli-cations in calculating forces acting on submergedobjects In general both hydrostatic and dynamic

ity and

Lines of constant density(increasing downward)

tilted


pressures must be considered but initially we look athydrostatic forces onlyTo start consider a submerged rectangular planar

surface oriented at an angle with respect to horizon-tal in a fluid as shown in Figure 6 (note that thesurface is shown in an oblique view) To simplifythe discussion it is assumed that the fluid has con-stant density although the general approach here iseasily extended to conditions of varying densityAs discussed above the pressure at any depth h isphfrac14 gh which acts perpendicularly on the surfaceindicated on the projection of the surface to the rightin Figure 6 The differential force acting on a smallarea element of the surface dA is dFfrac14 pdAfrac14 ghBdh(sin )1 where B is the width of the surfaceNote that dA is oriented horizontally and dh isassumed to be small enough that pressure may beassumed to be approximately constant over the areadA The total pressure force is then obtained by inte-grating between the limits h1 and h2

F frac14 B

sin

1

2ethh22 h21THORN frac1212

This force may be decomposed into horizontal andvertical components

Fx frac14 ethTHORNF sin frac14 B

2ethh22 h21THORN

Fz frac14 ethTHORNFcos frac14 B

2ethh22 h21THORNcos frac1213

where the negative signs indicate forces in the negativecoordinate directions It is useful to note that thepressure at the area centroid of the surface is

phc frac14 h1 thorn h2

2frac1214

where phc is the (hydrostatic) pressure at the areacentroid If this pressure acted on the entire area ofthe surface given by B(h2h1)(sin )

1 the total forcewould be ident ical to the result of eqn [12] In fact

dh

Dept

dA

B

Submergedsurface

(oblique view)

Figure 6 Hydrostatic forces acting on a submerged planar surface

it can be shown that this is a general result that thetotal pressure force on a submerged planar surfacemay be calculated as the product of the area andthe pressure evaluated at the area centroid It alsocan be shown (see lsquoFurther Readingrsquo) that the locationof the resultant pressure force acts through the cen-troid of the pressure prism defined by the pressuredistribution Similarly using moment balances it canbe shown that the resultant vertical force acting ona horizontal surface passes through the centroid ofthat area

For submerged curved surfaces consider forcesacting on the control volume designated by thecross-hatched area shown in Figure 7 Applying aforce balance in the horizontal direction gives

XFx frac14 0 ) Fsx frac14 B


where Fsx is the force of the curved surface acting onthe control volume in the x direction (pointing to theright) B is the width of the surface into the page andthe right-hand- side of eqn [15] is press ure forceacting on the right side of the control volume givenby eqn [12] In the vert ical direction the forcebalance givesX

Fz frac14 0 ) Fsz frac14 h1Bx thorn Vf frac1216

where Fsz is the force of the curved surface acting onthe control volume in the z direction (assumed topoint upwards) x is the length of the curved seg-ment projected in the vertical direction and Vf is thecross-hatched volume The first term on the right-hand -side of eqn [16] is the pressu re force acti ng onthe top surface of the control volume and the secondterm is the weight of fluid in the control volumeEqua tions [15] and [16 ] show that the forces actingon a curved surface may be calculated by consideringthe projected areas of the surface in the horizontaland vertical directions In other words the shape of

q

Pressuredistribution

h

h1

h2

h

Fsz

Fsx h2

h1

Arbitrary curvedsurface

Resultant vertical and horizontal pressure forcesacting on planar surfaces

Net forces in x and ydirections exerted by

curved surface oncontrol volume

Δx

Figure 7 Pressure forces acting on a curved surface

V

Projected area of object in vertical direction (this represents a surface on a horizontal plane)

W

F1

F2

Vf

Figure 8 Object submerged in a fluid of constant density


the surface does not matter except in so far as itdetermines the control volume (cross-hatched areain Figure 7)

Buoyancy

Buoyancy is directly related to hydrostatic pressureand may be considered as the net upward verticalforce due to pressure acting on a submerged objectThis result can be seen by considering an arbitrarilyshaped object with volume V submerged in a fluid ofconstant density r (Figure 8) A surface is drawnbelow the object to illustrate a space defined by verti-cal sides drawn everywhere tangential to the object(this surface is the area of the object projected verti-cally) A volume of fluid Vf is contained within thissurface and above the object The forces acting on theobject are the forces of fluid acting on its upper andlower surfaces F1 and F2 respectively and theobjectrsquos weight W There are no net forces in thehorizontal direction because there are no horizontal

variations in pressure It is easily seen (by applying aforce balance to the fluid above the object for exam-ple) that the force F1 is simply the weight of fluidsitting above the object occupying volume Vf Theforce F2 acting on the lower surface is the same as theweight of fluid that would have occupied the volume(VthornVf) ndash this can be seen by thinking of a columnof fluid with a surface drawn around the lower half ofthe object and applying the same sort of force balanceas above and noting that pressure is the same upwardsas downwards The force balance for the object in thevertical direction is then

Vf thornW frac14 ethV thorn Vf THORN ) W frac14 V frac1217This result shows that the weight of the object isbalanced by the weight of fluid that has been dis-placed by the object The displaced weight of fluid iscalled the buoyancy force It should be emphasizedthat the depth at which the object rests is arbitrary(unless the density of the ambient fluid varies) as is itsshape The most important consideration here is thedisplaced volume If the average density of the object(total mass divided by V) is equal to the density of thefluid the object will be in equilibrium at any locationin the water column If the objectrsquos density is greaterthan that of the fluid it will sink until it hits thebottom If the density is less than that of the fluidthe object will float at the surface with the degree ofsubmersion depending on the relative difference indensities and the displaced volume Sometimes refer-ence is made to the lsquosubmerged weightrsquo of an objectwhich is simply the actual weight minus the weight ofdisplaced fluid in which the object is placed Buoy-ancy is what allows ships which are typically made ofmaterials much denser than water to float The shipsimply has to be designed so that the weight of dis-placed water is greater than the weight of the ship andall its contents


Applying a force balance to fluid elements in a fluidwith density stratification gives rise to the concept oflsquorelative buoyancyrsquo where the density of the fluidelement is not the same as the density of its surround-ings There is a net reduced effect of gravity since theweight of the fluid element is partially offset by buoy-ancy For analysis of these situations the net effectivegravity is referred to as reduced gravity or relativebuoyancy defined by

g 0 frac14 g

0frac1218

where Dr is the difference between the density of theelement and its surroundings and r0 is a referencedensity value (usually taken as that of the surroundingfluid) Reduced gravity appears in problems associatedwith density-stratified flows and may take positive ornegative values depending on the sign of Dr

Dynamic Pressure

As previously noted dynamic pressure is associa-ted with fluid motion The simplest illustration ofdynamic pressure is obtained by considering theBernoulli equation which for cases of steady flowconstant density and negligible frictional losses maybe written as

2

2gthorn p

thorn z frac14 H frac12 19

where v is the fluid velocity andH is a constant knownas the Bernoulli constant or total head and is given byconditions of the problem (ie eqn [19] states that totalhead is constant for a given set of flow conditions)Each of the additive terms in eqn [19] has units of

Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)

Figure 9 Definition of head terms for Bernoulli equation (eqn [19])

and pressure heads is equal to the elevation at the water surface the

The energy line (EL) represents the elevation of total head in an enerthere is energy loss as may be induced by friction the EL slopes do

length The first term is known as the velocity headthe second is the pressure head and the third is theelevation head or simply elevationDefinitions of theseterms as well as concepts of hydraulic grade line(HGL) and energy line (EL) for the case of open chan-nel flow are shown in Figure 9 It may be noted that theBernoulli equation represents a statement of conserva-tion of energy where H represents the total energy ofthe flow in units of length or head In terms of realenergy units eqn [19] would be multiplied by massand by g The EL is a graphical representation of themagnitude of H so in a system where energy is con-served the EL or magnitude ofH is at a constant levelwhenmoving from one location to another in the flowIn other words considering two sections in the flow asin Figure 9 the total head should be the same at bothsections H1frac14H2 (note that velocity is assumed to beuniform at each cross section in this example ndash moredetailed discussion is needed when velocity gradientsare considered) The Bernoulli equation is developedfor comparison of conditions at difference points alonga common streamline or in the case of irrotationalflow as is usually assumed for open channel flow forany two points in the flow field By considering acase where velocities are zero everywhere eqn [19]reduces to a statement of hydrostatic pressure where(pg 1 thorn z) is a constant This sum is known as piezo-metric head and constancy of piezometric head (ieconstant position of the HGL) in a static fluid is easilyseen to be consistent with eqn [7]

In a moving fluid there is a sort of inverse relation-ship between velocity and pressure as indicated ineqn [19] That is region s of higher veloc ity genera llyhave lower pressures and vice-versa This is the maineffect for example that produces lift in airfoils andallows aircraft to fly ndash airfoils are designed so that

HGL

ELSection 2

z2

p2g

v222g

For hydrostatic pressure variations the sum of the elevation

position of which is also known as the hydraulic grade line (HGL)

gy-conserving system the EL is horizontal but in cases wherewnward in the direction of flow (shown as a dashed line)

z1= z2

(v2= 0)

2v1

1

Figure 10 Dynamic pressure force acting on planar surface inmoving stream of water point 2 is a stagnation point

h2h1

2

1

v1v2

Figure 11 Forces acting on a sluice gate in (two-dimensional)

open channel flow


there is a faster flow of air over the top of the airfoilthan over the bottom resulting in lower pressure onthe top than on the bottom with a net upward forceresulting For applications in water flow a typicalproblem might involve calculating the pressure forceacting on an object submerged in a flow A simplesituation of this type is illustrated in Figure 10 wherea flat plate is placed perpendicular to a moving streamof water At point 1 the velocity is v1 the pressureis p1 and the elevation is z1 At point 2 which isat the surface of the plate the velocity is (ideally)zero while the elevation z2frac14 z1 Applying theBernou lli equ ation [19] then gives

p2 frac14 p1 thorn 1

2 21 frac12 20

Point 2 is known as a stagnation point which isdefined anywhere where the velocity is zero and thesecond term on the right-hand-side of eqn [20] isthe dynamic pressure component In this case thedynamic pressure acting at point 2 attains the highestvalue possible since point 2 is a stagnation point andv2frac14 0 (any velocity v2gt 0 would reduce p2 by anamount p2

2=2) When comparing pressures at twopoints in a fluid any difference due to different velo-cities comes from a dynamic pressure effect whichdepends on the difference in velocities squared (v2)In general to calculate the total pressure force acting

on a submerged object would require an integrationof the pressure distribution on the surface of the objectwhich as described previously would require detailedknowledge of the velocity distribution Fortunatelyin many cases a simpler approach may be appliedbased on general force balance and continuity consid-erations For example consider calculations of forceacting on a sluice gate as shown in Figure 11 Againassuming constant velocities at each cross section(1 and 2) an integral momentum equation may beapplied along with the continuity and Bernoulli equa-tions to solve for the net hydraulic (pressure) forceacting on the gate Considering a control volume con-sisting of the water between sections 1 and 2 andassuming steady flow continuity states that flow rateof water entering the control section must be the sameas that leaving so

1h1 frac14 2h2 frac1221where unit width has been assumed (ie two-dimensional flow is considered for this problem)Using the channel bed as datum and neglecting headloss the Bernoulli equation gives

212g

thorn h1 frac14 222g

thorn h2 frac1222

where h is the piezometric head If the depths h1 and h2are known eqns [21] and [22] can be used to find

the velocities at each section and therefore theflow in the channel Applying the Reynolds TransportTheorem to evaluate forces on the control volumegives the integral momentum equation

pc1h1 pc2h2 Fg frac14 qeth2 1THORN frac1223where pc is the (hydrostatic) pressure evaluated at thecentroid of each cross section (recall previous discus-sion of forces on submerged surfaces) Fg is the totalforce exerted by the gate on the fluid in the controlvolume assumed to act in the negative x direction andqfrac14 vh is the two-dimensional flow rate or flow perunit width The force Fg is the net integrated effect ofthe pressure distribution on the gate resulting fromboth hydrostatic and dynamic components and isfound using eqn [23] without needing to actually cal-culate the pressures directly Thus with a simplemeasurement of depths upstream and downstreamof the gate the net force on the gate is found wherethe force on the gate is in the opposite direction as theforce found from eqn [23]

Pressure in the Equations of Motion

In several of the above examples it has been implicitlyassumed that the pressure variation was approxi-mately hydrostatic even in the case where velocitywas not zero This assumption is evaluated in this


section which explores the impact of pressure differ-ences on the equations of motion as would be used indeveloping mathematical models of flows and circu-lation for environmental analyses in lakes andstreams The equations governing fluid flow consistof statements of conservation of mass (continuity)momentum and energy Of particular interest hereare the momentum equations or NavierndashStokesequations which in component form are written as

u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p

x g thorn r2w frac1226

where f is the Coriolis parameter defined as twice thedaily rotation rate of the earth times the sine of thelatitude and n is kinematic viscosity For inlandwatersexcept for very large lakes such as the LaurentianGreatLakes of North America the Coriolis term may beneglected Also it is easily seen that in the case ofno motion u frac14 v frac14 w frac14 0 eqns [24]ndash[26] reduce toeqn [4] The main hydrostatic pressure equation refersto the vertical distribution of pressure In most cases ofnatural flows the motions are predominantly in hori-zontal directions sow is small as are vertical accelera-tions so that all terms in eqn [26] are negligible exceptfor the first two terms on the right-hand-side consis-tent with hydrostatic pressure variation in the verticaldirection There are certain situations where this is notthe case such as during fall or spring overturns in lakesbut these situations are generally of limited temporalduration It should be noted that assuming a hydro-static pressure variation in the vertical direction doesnot necessarily imply any assumption for horizontalpressure gradientsFor applications in model development for inland

waters it is useful to explore the impact of the pres-sure term in the NavierndashStokes equations Here con-sider the v ector form of eqns [24]ndash[ 26]

~v

tthorn~v r~v thorn 2~~v frac14~g 1

rp thorn vr2~v frac1227

where ~ is the earthrsquos rotation vector The pressureterm as discussed previously may be considered asthe sum of hydrostatic and dynamic componentswhere the hydrostatic part may be written as

ph frac14 pr Zz

zr

gdz frac1228

where pr is a reference value at zfrac14 zr Note that eqn[28] is simply the integrated form of the last part

of eqn [4] Lett ing p frac14 ph thorn pd wher e pd is thedynami c press ure and substitut ing eqn [28 ] the pres-sure gradi ent term in eqn [27] may be written as

1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr

rdz grzthorn grrzr thorn 1

rpd frac1229

Then substitut ing eqn [29] into eqn [27] an d ap ply-ing the Boussi nesq approxim ation (neglect de nsityvariations except in the buoyancy term) the result is

~v

tthorn~vr~v thorn 2~~v frac14 1

0rethpr thorn pd THORN

thorn g

0

Zz

zr

rdz grzr thorn vr2~v frac1230

where r0 is a reference density value usually thedensity at 4 C in freshwater systems On the right-hand -side of eqn [30] the grad ient of refe rence pres-sure pr may usually be neglected The second termis the effect of density variations which are importantfor stratified fluids and the third term is the effect ofreferenc e surfa ce gradien ts (such as waves) Alo ngwith continu ity an d energy equati ons eqn [30]may serve as a general starting point for developingmodels of fluid motion in natural waters although inmany instances it is possible to neglect some of theterms and use a simplified form of the equation

Biological Responses

The above discussion focuses on the physical descrip-tion of pressure how it varies in a fluid how forcesare manifested on submerged objects and how it isincorporated in the equations that would form thebasis of hydrodynamic and water quality models ofinland waters Other considerations apply to variousspecies that live or play in water and examples ofissues related to algae fish and humans are presentedbriefly here

For submerged objects buoyancy is the main forceof interest As shown previously buoyancy is thenet result of pressure forces in the vertical directionPressures in horizontal directions or at least in thedirection of movement are of interest in determiningdrag that must be overcome to maintain such move-ment The simplest biological response and move-ment in the water column is achieved through theprocess of buoyancy regulation which is used bycertain species of algae to position themselves opti-mally in regions of preferable temperature light and


nutrient availability These algae are mostly of theblue-green type (or cyanobacteria) which also cancause nuisance and even harmful (toxic) bloomsBuoyancy regulation is achieved by increasing ordecreasing gas volume in vesicles in the cells Byincreasing or decreasing volume the displaced waterand hence the buoyancy force acting on the cell isaltered Increasing volume increases the buoyancyforce and causes the cell to rise and up to five-foldvariations in rising or falling rates have been observedThe actual rate of rise depends on the density struc-ture of the water column in which the organismfloats since buoyancy is the weight of the displacedwater which changes as a function of densityFor fish the physical interactions with the water

environment are more complicated due to locomo-tion Some fish make use of a gas-filled cavity called aswim bladder or gas bladder to maintain buoyancyand stability Additional uplift forces can be obtainedby swimming through the same Bernoulli effectnoted above that allows airplanes to fly Howeverthis dynamic lift is achieved only when there is for-ward motion With respect to the swim bladder inorder to maintain a constant buoyancy the volume ofthe bladder must remain approximately constant asthe fish swims in different depths where pressurechanges and this requires some interesting physiologi-cal responses Near the surface the pressure inthe water is close to atmospheric pressure but aspreviously described pressure increases by about1 atm for every 10m of depth Unlike water air iscompressible and volume decreases as pressure in-creases so there is a tendency for reduced buoyancyat greater depths In order to maintain neutral buoy-ancy within a water column an effort must be madeby the fish with swim bladders to keep the volumeof their swim bladder constant Methods of maintain-ing some lsquohydrostatic equilibriumrsquo varies among dif-ferent groups and species of fish This maintenanceis usually done by slight secretions or resorptions ofgas within the swim bladder itself or by releaseof gas through a duct lsquoPhysoclistsrsquo are fish (eg theperch Perca fluviatilus) that have special gas secret-ing and resorbing sites on the swim bladder wall thatlet the fish descend or ascend respectively lsquoPhyso-stomesrsquo are fish (eg the eel Anguilla anguilla) thathave a pneumatic duct that extends from the swimbladder to the esophagus Such a duct allows thesefish to release or lsquoburprsquo some expanding gas as thefish ascends Still other fish (eg castor oil fishRuvettus pretiosus) are able to maintain neutralbuoyancy through alterations in their quantities oflipid storage Interestingly fish like tuna (eg the

little Pacific mackerel tuna Euthynmus affinus) andsharks have no swim bladders This latter group aswell as species like dolphins gain hydrodynamic liftby their shape but they must swim continuously tokeep from sinking

Perhaps of more direct interest for humans is theattention one must pay to pressure when diving As adiver moves into deeper or shallower water the pres-sure changes and affects the balance between concen-trations of gases in the dissolved (liquid) and gaseousphases following Henryrsquos Law This law expressesthe equilibrium between the dissolved phase concen-tration of a gas and its pressure in the surroundingsIn essence as outside pressure increases gases arelsquopushedrsquo into the dissolved phase The problem fordivers occurs when they ascend too quickly followinga deep dive As pressure is reduced gases move intothe gaseous phase and if pressure is reduced toorapidly the gas cannot leave the blood stream quicklyenough and gas bubbles mostly nitrogen form in theblood In other words the re-dissolution process doesnot have enough time to accommodate the gassesmoving out of solution due to the pressure changeThis situation leads to lsquothe bendsrsquo also known asdecompression or caisson sickness (the latter defini-tion comes from the situation where workers wouldwork in pressurized caissons or boxes lowered instreams for construction of structures such as bridgetowers ndash the interior of the caisson was pressurized toequal that of the surrounding water to prevent waterfrom entering the work area and workers leavingthe pressurized area too quickly would suffer thesame symptoms as divers who ascended too quicklyfrom a dive this was a significant problem in thebuilding of the Brooklyn Bridge for example) Symp-toms of lsquothe bendsrsquo include pain in the joints musclecramps sensory system failure and in extreme caseseven death

While the material in this article goes into somedepth with regard to the scientific and engineeringanalysis of pressure it is important to recognize thatdifferent species have different responses to pressurevariations The above examples represent only asmall sampling of these reactions and how pressureand its net vertical force buoyancy is important inregulating our physical environment

Further Reading

Batchelor GK (1967) An Introduction to Fluid DynamicsCambridge Cambridge University Press

Morris HM and Wiggert JM (1972) Applied Hydraulics inEngineering 2nd edn New York Wiley


Munson BR Young DF and Okiishi TH (1998) Fundamentals ofFluid Mechamics 3rd edn New York Wiley

Pelster B (1998) Buoyancy In Evans DH and Boca R (eds) ThePhysiology of Fishes 2nd edn Boca Raton FL CRC Press

Rubin H and Atkinson J (2001) Environmental Fluid MechanicsNew York Marcel Dekker Inc

Shames IH (2003) Mechanics of Fluids 4th edn New York

McGraw-Hill

Turner JS (1973) Buoyancy Effects in Fluids Cambridge

Cambridge University Press

Relevant Websites

httppadicom ndash Diving information and diving tableshttpwwwamericandivecentercomdeeppreviewpd04htm ndash Diving

information and diving tables

httphyperphysicsphy-astrgsueduHbosepmanhtml ndash General dis-

cussion of water pressurehttpwwwatozdivingconzwaterpressurehtm ndash Calculator for

pressure in salt water

Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites


x dire ction side s 3 and 4 refe rring to side s withperpendi culars along the y direction and sides 5 and6 with perpendicu lars a long the z dire ction Note thatthe z dire ction is orien ted paral lel to the dire ction ofgravity but points positive upwar ds If ph is con sid-ered to be the press ure at the cente r of the elementthen press ures along each of the faces can beexpres sed in terms of ph using a trunca ted TaylorSeries exp ansion an d force (product of stress a ndarea on which it acts) balanc es in each of the threecoordi nate directio ns may be written as

ethx THORN ph1 ethd y dz THORN frac14 ph2 ethd y d z THORN ) ph ph x

d x

2

frac14 ph thorn ph x

d x

2

frac12 1

eth y THORN ph3 ethd x dz THORN frac14 ph4 ethd x d z THORN ) ph ph y

d y

2

frac14 ph thorn ph y

d y

2

frac12 2

ethz THORN ph5 ethd x d y THORN frac14 ph6 ethd x dy THORN thorn g eth dx d y d z THORN ) ph ph z

d z

2

frac14 ph thorn ph z

d z

2

thorn g d z frac12 3

dx

dy

Normal stress(pressure)

Shear stresses(in plane of surface)

dz

Figure 1 Stresses acting on one surface of a small fluid

element

3

1y

x

zh

We

Figure 2 Forces acting on a small fluid element under static conditioby the rectangular surface and the pressure acting on each of the fac

where ph i is the hyd rostatic press ure actin g on face i gis gravitational acceleration and r is the fluid densityNote that dx dy and dz are assumed to be smallenough that any variations in phi over the area ofthe side may be neglecte d Equa tion [3] differs fromeqns [1] and [2] only in that the fluid weight isinclud ed in the force balanc e Simp lifying eqns[1]ndash[3 ] resul ts in

phx

frac14 phy

frac14 0phz

frac14 g frac14 frac124

where g frac14 rg is the specific weight of the fluidKeeping the assumption of constant density and

letting the fluid be water the pressure on the top ofthe fluid element in Figure 2 is given by integratingthe last part of eqn [4] resulting in p frac14 p0thorn ghwhere p0 is the pressure at the water surface andhfrac14 z0z is the water depth where z0 is the elevationat the water surface and z is the elevation at the top ofthe element Also note that the weight (W) of thewater column above the fluid element is Wfrac14 ghAwhere Afrac14 dxdy is the area of the element If p0frac14 0is assumed then the pressure acting on the topsurface of the fluid element in Figure 2 is simply theweight of water sitting above the element divided bythe area (dxdy) Using similar reasoning the pressureat the water surface is actually given by the weightof air above the surface per unit area (note thatpressure can be defined in terms of weight whetheror not the density is constant but if density is notconstant the weight must be determined by integra-tion of the density distribution) In fact Newton longago deduced that air has mass by determining thatwater could be pumped up a tube by extracting airat the top only to a maximum height of about 33 ft(10m) since the weight of a column of water33 ft high weighs approximately the same as a similarcolumn of air stretching through the entire

5

6

4

2 Pressure

Water surface

ight

ns (no fluid motion) forces include the weight of the fluid enclosedes (six totally) of the element






ph1

ph2Δph

z1

z2

z


Δz



phz




phx

frac14 x frac127





phr

frac14 r2 frac128


Density Variations


ph2 ph1 frac14 gk


frac129



Fluid element



hDirection of

pressure gradient










ity and


tilted




F frac14 B

sin

1




2ethh22 h21THORN





2frac1214



dh

Dept

dA

B

Submergedsurface

(oblique view)









q


h

h1

h2

h

Fsz

Fsx h2

h1





Δx


V


W

F1

F2

Vf




Buoyancy






g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites






ph1

ph2Δph

z1

z2

z


Δz



phz




phx

frac14 x frac127





phr

frac14 r2 frac128


Density Variations


ph2 ph1 frac14 gk


frac129



Fluid element



hDirection of

pressure gradient










ity and


tilted




F frac14 B

sin

1




2ethh22 h21THORN





2frac1214



dh

Dept

dA

B

Submergedsurface

(oblique view)









q


h

h1

h2

h

Fsz

Fsx h2

h1





Δx


V


W

F1

F2

Vf




Buoyancy






g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites



phr

frac14 r2 frac128


Density Variations


ph2 ph1 frac14 gk


frac129



Fluid element



hDirection of

pressure gradient










ity and


tilted




F frac14 B

sin

1




2ethh22 h21THORN





2frac1214



dh

Dept

dA

B

Submergedsurface

(oblique view)









q


h

h1

h2

h

Fsz

Fsx h2

h1





Δx


V


W

F1

F2

Vf




Buoyancy






g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites




F frac14 B

sin

1




2ethh22 h21THORN





2frac1214



dh

Dept

dA

B

Submergedsurface

(oblique view)









q


h

h1

h2

h

Fsz

Fsx h2

h1





Δx


V


W

F1

F2

Vf




Buoyancy






g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites

Fsz

Fsx h2

h1





Δx


V


W

F1

F2

Vf




Buoyancy






g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites



g 0 frac14 g

0frac1218


Dynamic Pressure


2

2gthorn p



Section 1

Flow

z

p1g

v122g

H

z = 0 (datum)






HGL

ELSection 2

z2

p2g

v222g




z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites

z1= z2

(v2= 0)

2v1

1


h2h1

2

1

v1v2


open channel flow




2 21 frac12 20





212g


thorn h2 frac1222








u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites



u

tthorn u

u

xthorn v

u

ythorn w

u

z fv frac14 1

p

xthorn r2u frac1224

v

tthorn u

v

xthorn v

v

ythorn w

v

zthorn fu frac14 1

p

ythorn r2v frac1225

w

tthorn u

w

xthorn v

w

ythorn w

w

zfrac14 1

p




~v




ph frac14 pr Zz

zr

gdz frac1228



1

rp frac14 1

rpr g

r

Zz

zr

dzthorn 1

rpd frac14 1

rpr

g

Zz

zr


rpd frac1229


~v



thorn g

0

Zz

zr












Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites







Further Reading








McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites






McGraw-Hill



Relevant Websites






Pressure

Introduction


Density Variations


Buoyancy

Dynamic Pressure



Further Reading

Relevant Websites

encyclopedia of inland waters || pressure

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