compare to Mannings equation

Transcription

1 330 Fluid dynamics Density and viscosity help to control velocity and shear in fluids Density ρ (rho) of water is about 700 times greater than air (20 degrees C) Viscosity of water about 55 times greater than air (20 degrees C) Both viscosity and density increase with decreasing temperature Two principal flow types: laminar and turbulent Laminar flow = unidirectional o Typically streamlines are parallel, velocity is constant o Mudflows and ice are typically laminar Turbulent flow = polydirectional, with a dominant average o Turbulence (intertwining of streamlines) influenced by Increase in velocity or a decrease in viscosity o Velocity averages out over time, but is varies from instant to instant o Eddies are highly turbulent movements o Upwards-directed flowlines slow settling grains o Turbulent fluids have an apparent higher viscosity than laminar fluids because they resist distortion better Molecular viscosity µ (mu) and Shear stress τ (tau) Measure of resistance to change in shape at finite speeds of flow. Defined as the ratio of shear stress τ (tau) to rate of deformation across a fluid with depth o τ = µ (du/dy) and µ = τ /(du/dy) o du = change in velocity o dy = change in height Kinematic viscosity or the ratio of the rate of resistance to change in shape (i.e. molecular viscosity) to density o ν = µ / ρ is related to fluid turbulence

2 Reynolds Number (1833) Osborne Reynolds, professor of engineering, Univ. of Manchester Ratio of inertial force to viscous force o Inertial force causes turbulence o Viscous force reduces turbulence or With o v s - mean fluid velocity o L - characteristic length (equal to diameter 2r if a cross-section is circular), o µ - (absolute) dynamic fluid viscosity (molecular (dynamic) viscosity) o ν (nu) - kinematic fluid viscosity: ν = µ / ρ, o ρ - fluid density. R e is dimensionless If R e = low, flow tends to be laminar o Viscous forces dominate o Low flow velocity o And/or shallow depth If R e = high, flow tends to be turbulent Critical value for laminar to turbulent transition = 500 to 2000 Dependent on boundary conditions (e.g. channel geometry and depth) Useful for comparing systems that vary by magnitude of scale (e.g. models) Boundary layers Molecular adhesion (velocity = 0, polarity of water forms a buffering layer) Viscous or laminar sublayer Subsequent layers may be either laminar or turbulent, and vary in thickness for each system Hydraulic smooth boundaries have bed roughness less than the sized of the viscous sublayer Hydraulically rough boundaries have bed roughness greater than the sized of the viscous sublayer Reason why mud often remains behind even in rapid flow if confined to viscous sublayer. Boundary shear stress (Allen 1994) τ 0 = γρghs o where tau = force per unit area parallel to the bed o γ = density of fluid?; ρ = fluid density? ;g = gravitational acceleration o h = flow depth ; and s = slope gradient τ b = ρ w grs bed shear strength compare to Mannings equation

3 Manning s (1891) Equation with ν = velocity of fluvial system η = roughness of channel S = channel slope (dimensionless) R = hydraulic radius = A/WP = ratio of the x-sectional area to its wetted perimeter 1.49 in the original paper is designated as k, and is for unit conversion in English units. It is not for metric units. Robert Manning ( ) developed formula in 1891 paper On the Flow of Water in Open Channels and Pipes," published in Transactions of the Institution of Civil Engineers of Ireland. He investigated the variable factors that influence fluid velocity and to quote " by taking the mean results of all of them an approximation to the truth might be arrived at." Manning was an accountant for an engineering firm, and had no formal training in fluid mechanics or engineering. He was self-taught. To quote Manning on formulae: And now a few words, again addressed to the younger members, with regard to the use of formulae. I very much fear that if I were to illustrate any observations I have to make with chalk on the black-board, a dozen note-books would be taken out, the formula copied without investigation, probably to my discredit, and eventually worked to death. It should be remembered that a formula is only a short memorandum (put in a shape fit for ready use) of the result arrived at after a patient consideration of the facts and principles upon which it is founded, and to use it without investigation is the merest empiricism.

4 Froude number u = mean flow velocity, g = acceleration due to gravity h = a representative length scale (e.g. water depth) F r is dimensionless, useful for scaling o If F r is less than 1, flow is deemed subcritical, tranquil, or streaming. Sedimentologically, this is refered to as the Lower Flow Regime (Simons and Richardson, 1961). Waveforms on surface of flow are out-of-phase with bedforms, and bedforms migrate downstream. o If F r is greater than 1, flow is supercritical, rapid, or shooting. Sedimentologically, this is refered to as the Upper Flow Regime (Simons and Richardson, 1961). Waveforms on surface of flow are in phase with bedforms, and bedforms migrate upstream. Includes planar bedforms in plane bed flow (lowly supercritical). Named after William Froude ( ), engineer, hydrodynamicist and naval architect. Formulated reliable laws on water resistance for ship design and stability. Hjulstrőm-Sundborg Diagram Experimentally derived threshold for entraining quartz grains on a plane bed in water 1 meter deep Log-log pattern Variables are mean fluid velocity and grain size Shows critical threshold Only works in freshwater streams (fluid and grain densities and dynamic viscosities are constant)

5 Shields Diagram Plots dimensionless shear stress (β) instead of flow velocity against the grain Reynolds number (R eg ). Useful for air flow as well as water and incorporates density into critical threshold for grain entrainment. Shields, A., Application of Similarity Principles and Turbulence Research to Bedload Movement. In: Ott, W.P. and Unchelen, J.C. (Translators), Mitteilungen der preussischen Nersuchsanstalt fur Wassed Brau end Schiffbauer. Report 167, California Institute of Technology, Pasedena, California. Dimensionless shear stress o β = τ t / ((ρ d ρ f )gd) o with τ t = boundary shear stress o ρ d = particle density o ρ = fluid density o g = gravitational acceleration o D = particle diameter o An increase in β is either an increase in shear stress, or an decrease in the difference between particle and fluid density, or an decrease in grain size. Grain Reynolds number (R eg ) o R eg = UD/ν o With U = shear velocity (includes turbulence) o D = grain diameter o And ν = kinematic viscosity o An increase in R eg is either an increase in turbulence or shear velocity, grain diameter, or a decrease in viscosity

6 Stokes (1845) Law Final settling velocity of quartz spheres V= (D 2 )(( ρ d ρ f )g/(18µ)) o D = sphere diameter o ρ d = particle density o ρ f = fluid density o g = gravitational acceleration o µ = fluid viscosity Experimental verification for particles less than about.2 mm diameter (fine to very fine sand. Larger particle calculated velocities are often too fast (deviation from spherical shape significant, and increased turbulence as suggested by Reynolds numbers). This means that it cannot be used accurately for sand, a very common sediment!!! Concentration of suspended sediment also alters accuracy it increases apparent viscosity and density, and also grain impacts. George Gabriel Stokes, ( ), professor at Cambridge, perhaps most brilliant part of the trio of mathematical physicists who made Cambridge famous (James Clerk Maxwell and William Thomson (Lord Kelvin)). Published on fluid dynamics in early career, later focusing on waves and their behavior in and through various media (e.g. sound, light etc.).