Joseph S. Walder
2016
<div class="NLM_sec NLM_sec_level_1 hlFld-Abstract"><p>A method of achieving a dimensionless collapse of erosion-rate data for cohesive sediments is proposed and shown to work well for data collected in flume-erosion tests on mixtures of sand and mud (silt plus clay sized particles) for a wide range of mud fraction. The data collapse corresponds to a dimensional erosion law of the form <span class="equationTd"><span id="MathJax-Element-1-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><mi>E</mi><mo>&#x223C;</mo><msup><mrow><mo stretchy="false">(</mo><mi>&#x3C4;</mi><mo>&#x2212;</mo><msub><mrow><mi>&#x3C4;</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup></mrow></math>"><span id="MathJax-Span-1" class="math"><span><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="mrow"><span id="MathJax-Span-4" class="mi">E</span><span id="MathJax-Span-5" class="mo">∼</span><span id="MathJax-Span-6" class="msup"><span id="MathJax-Span-7" class="mrow"><span id="MathJax-Span-8" class="mo">(</span><span id="MathJax-Span-9" class="mi">τ</span><span id="MathJax-Span-10" class="mo">−</span><span id="MathJax-Span-11" class="msub"><span id="MathJax-Span-12" class="mrow"><span id="MathJax-Span-13" class="mi">τ</span></span><span id="MathJax-Span-14" class="mrow"><span id="MathJax-Span-15" class="mi">c</span></span></span><span id="MathJax-Span-16" class="mo">)</span></span><span id="MathJax-Span-17" class="mrow"><span id="MathJax-Span-18" class="mi">m</span></span></span></span></span></span></span><span class="MJX_Assistive_MathML">E∼(τ−τc)m</span></span></span>, where <span class="equationTd"><span id="MathJax-Element-2-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mi>E</mi></math>"><span id="MathJax-Span-19" class="math"><span><span id="MathJax-Span-20" class="mrow"><span id="MathJax-Span-21" class="mi">E</span></span></span></span><span class="MJX_Assistive_MathML">E</span></span></span> is erosion rate, <span class="equationTd"><span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mi>&#x3C4;</mi></math>"><span id="MathJax-Span-22" class="math"><span><span id="MathJax-Span-23" class="mrow"><span id="MathJax-Span-24" class="mi">τ</span></span></span></span><span class="MJX_Assistive_MathML">τ</span></span></span> is shear stress, <span class="equationTd"><span id="MathJax-Element-4-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><msub><mi>&#x3C4;</mi><mi>c</mi></msub></mrow></math>"><span id="MathJax-Span-25" class="math"><span><span id="MathJax-Span-26" class="mrow"><span id="MathJax-Span-27" class="mrow"><span id="MathJax-Span-28" class="msub"><span id="MathJax-Span-29" class="mi">τ</span><span id="MathJax-Span-30" class="mi">c</span></span></span></span></span></span><span class="MJX_Assistive_MathML">τc</span></span></span> is the threshold shear stress for erosion to occur, and <span class="equationTd"><span id="MathJax-Element-5-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><mi>m</mi><mo>&#x2248;</mo><mn>7</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></math>"><span id="MathJax-Span-31" class="math"><span><span id="MathJax-Span-32" class="mrow"><span id="MathJax-Span-33" class="mrow"><span id="MathJax-Span-34" class="mi">m</span><span id="MathJax-Span-35" class="mo">≈</span><span id="MathJax-Span-36" class="mn">7</span><span id="MathJax-Span-37" class="mo">/</span><span id="MathJax-Span-38" class="mn">4</span></span></span></span></span><span class="MJX_Assistive_MathML">m≈7/4</span></span></span>. This result contrasts with the commonly assumed linear erosion law <span class="equationTd"><span id="MathJax-Element-6-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><mi>E</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>&#x3C4;</mi><mo>&#x2212;</mo><msub><mrow><mi>&#x3C4;</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math>"><span id="MathJax-Span-39" class="math"><span><span id="MathJax-Span-40" class="mrow"><span id="MathJax-Span-41" class="mrow"><span id="MathJax-Span-42" class="mi">E</span><span id="MathJax-Span-43" class="mo">=</span><span id="MathJax-Span-44" class="msub"><span id="MathJax-Span-45" class="mrow"><span id="MathJax-Span-46" class="mi">k</span></span><span id="MathJax-Span-47" class="mrow"><span id="MathJax-Span-48" class="mi">d</span></span></span><span id="MathJax-Span-49" class="mo">(</span><span id="MathJax-Span-50" class="mi">τ</span><span id="MathJax-Span-51" class="mo">−</span><span id="MathJax-Span-52" class="msub"><span id="MathJax-Span-53" class="mrow"><span id="MathJax-Span-54" class="mi">τ</span></span><span id="MathJax-Span-55" class="mrow"><span id="MathJax-Span-56" class="mi">c</span></span></span><span id="MathJax-Span-57" class="mo">)</span></span></span></span></span><span class="MJX_Assistive_MathML">E=kd(τ−τc)</span></span></span>, where <span class="equationTd"><span id="MathJax-Element-7-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><msub><mi>k</mi><mi>d</mi></msub></mrow></math>"><span id="MathJax-Span-58" class="math"><span><span id="MathJax-Span-59" class="mrow"><span id="MathJax-Span-60" class="mrow"><span id="MathJax-Span-61" class="msub"><span id="MathJax-Span-62" class="mi">k</span><span id="MathJax-Span-63" class="mi">d</span></span></span></span></span></span><span class="MJX_Assistive_MathML">kd</span></span></span> is a measure of how easily sediment is eroded. The data collapse prompts a re-examination of the way that results of the hole-erosion test (HET) and jet-erosion test (JET) are customarily analyzed, and also calls into question the meaningfulness not only of proposed empirical relationships between <span class="equationTd"><span id="MathJax-Element-8-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><msub><mi>k</mi><mi>d</mi></msub></mrow></math>"><span id="MathJax-Span-64" class="math"><span><span id="MathJax-Span-65" class="mrow"><span id="MathJax-Span-66" class="mrow"><span id="MathJax-Span-67" class="msub"><span id="MathJax-Span-68" class="mi">k</span><span id="MathJax-Span-69" class="mi">d</span></span></span></span></span></span><span class="MJX_Assistive_MathML">kd</span></span></span> and <span class="equationTd"><span id="MathJax-Element-9-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><msub><mi>&#x3C4;</mi><mi>c</mi></msub></mrow></math>"><span id="MathJax-Span-70" class="math"><span><span id="MathJax-Span-71" class="mrow"><span id="MathJax-Span-72" class="mrow"><span id="MathJax-Span-73" class="msub"><span id="MathJax-Span-74" class="mi">τ</span><span id="MathJax-Span-75" class="mi">c</span></span></span></span></span></span><span class="MJX_Assistive_MathML">τc</span></span></span>, but also of the erodibility parameter <span class="equationTd"><span id="MathJax-Element-10-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><msub><mi>k</mi><mi>d</mi></msub></mrow></math>"><span id="MathJax-Span-76" class="math"><span><span id="MathJax-Span-77" class="mrow"><span id="MathJax-Span-78" class="mrow"><span id="MathJax-Span-79" class="msub"><span id="MathJax-Span-80" class="mi">k</span><span id="MathJax-Span-81" class="mi">d</span></span></span></span></span></span><span class="MJX_Assistive_MathML">kd</span></span></span> itself. Fuller comparison of flume-erosion data with hole-erosion and jet-erosion data will require revised analyses of the HET and JET that drop the assumption <span class="equationTd"><span id="MathJax-Element-11-Frame" class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math>"><span id="MathJax-Span-82" class="math"><span><span id="MathJax-Span-83" class="mrow"><span id="MathJax-Span-84" class="mrow"><span id="MathJax-Span-85" class="mi">m</span><span id="MathJax-Span-86" class="mo">=</span><span id="MathJax-Span-87" class="mn">1</span></span></span></span></span><span class="MJX_Assistive_MathML">m=1</span></span></span> and, in the case of the JET, certain simplifying assumptions about the mechanics of jet scour.</p></div>
application/pdf
10.1061/(ASCE)HY.1943-7900.0001068
en
American Society of Civil Engineers
Dimensionless erosion laws for cohesive sediment
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