DYNAMIC EQUATIONS FOR STEADY SPATIALLY VARIED FLOW

BOTH MOMENTUM AND ENERGY PRINCIPLES CAN BE ADOPTED TO DERIVE DYNAMIC EQUATIONS DESCRIBING SPATIALLY VARIED FLOW IN CHANNELS. SUCH EQUATIONS WERE DERIVED FOR STEADY FLOW OF AN INCOMPRESSIBLE VISCOUS FLUID IN A UNIFORM CHANNEL WITH SPATIALLY VARIABLE LATERAL INFLOW OR OUTFLOW THROUGH A TWO-DIMENSIONAL APPROACH CONSIDERING THE CROSS-SECTIONAL VARIATIONS IN VELOCITY AND PRESSURE. BECAUSE OF THE BASIC DIFFERENCE IN PHYSICAL CONCEPTS OF THE MOMENTUM AND ENERGY PRINCIPLES, THE MATHEMATICAL EXPRESSIONS OBTAINED BASED ON THESE TWO PRINCIPLES ARE INHERENTLY DIFFERENT. THE MOMENTUM APPROACH IS A VECTOR RELATIONSHIP WHICH EQUATES THE NET MOMENTUM FLUX AND THE EXTERNAL FORCES ACTING ON A CONTROL VOLUME IN THE DIRECTION OF FLOW, WHILE THE ENERGY APPROACH IS A SCALAR RELATIONSHIP CONSIDERING NOT ONLY THE WORK DONE AND ENERGY TRANSFER THROUGH THE SURFACE BOUNDING THE CONTROL VOLUME BUT ALSO THE WORK DONE BY GRAVITY FORCE AND THE ENERGY DISSIPATION DUE TO INTERNAL STRESS WITHIN THE VOLUME. THE DYNAMIC EQUATIONS OBTAINED FROM THE TWO-DIMENSIONAL APPROACH CAN BE CONVERTED INTO ONE DIMENSIONAL FORM. IN ADDITION TO THE CONVENTIONALLY USED MOMENTUM FLUX CORRECTION FACTOR AND THE KINETIC ENERGY FLUX CORRECTION FACTOR, TWO ADDITIONAL CORRECTION FACTORS, "K" AND "NU", ARE INTRODUCED TO MAKE THE CONVERSION POSSIBLE IN THE MOMENTUM AND ENERGY EQUATIONS RESPECTIVELY. BOTH K AND NU ARE FUNCTIONS OF PRESSURE DISTRIBUTION AND THE CROSS-SECTIONAL SHAPE. IN ADDITION, NU IS ALSO A FUNCTION OF VELOCITY DISTRIBUTION. HOWEVER, THE MOST IMPORTANT DIFFERENCE IN THE DYNAMIC EQUATIONS DERIVED RESPECTIVELY FROM MOMENTUM AND ENERGY APPROACHES IS THAT BETWEEN THE FRICTION SLOPE AND THE DISSIPATED ENERGY GRADIENT. FOR A STEADY SPATIALLY VARIED FLOW THERE ARE SIX GRADIENTS INVOLVED, NAMELY, THE CHANNEL SLOPE, THE FREE SURFACE SLOPE, THE AVERAGE PIEZOMETRIC-HEAD GRADIENT, THE TOTAL HEAD GRADIENT, THE FRICTION SLOPE, AND THE DISSIPATED ENERGY GRADIENT. ONLY FOR STEADY UNIFORM FLOW, WITH THE LATERAL FLOW RATE PER UNIT LENGTH EQUAL TO ZERO, ARE THESE SIX GRADIENTS EQUAL. /AUTHOR/

• Supplemental Notes:
  • Vol 96, No HY 3, PROC PAPER 7179, PP 801-814, 1 FIG, 14 REF,
• Authors:
  • Yen, B C
  • Wenzel, H G
• Publication Date: 1970-3

Media Info

• Serial:
  • Journal of the Hydraulics Division
  • Publisher: American Society of Civil Engineers

Subject/Index Terms

• TRT Terms: Channel flow; Cross sections; Dissipation; Ditches; Dynamics; Energy; Equations; Flow; Fluxes; Friction; Functions (Mathematics); Heat flux; Momentum; Pressure; Standardization; Variables; Velocity
• Uncontrolled Terms: Pressure distribution; Steady flow
• Old TRIS Terms: Derivation; Flow channel; Variations; Velocity distribution
• Subject Areas: Energy; Highways; Hydraulics and Hydrology;

Filing Info

• Accession Number: 00204279
• Record Type: Publication
• Files: TRIS
• Created Date: May 19 1970 12:00AM