<p>Helically coiled reactors (HCRs) are increasingly employed in industrial applications due to their enhanced heat and mass transfer characteristics, compact design and improved mixing performance compared to straight tube systems. Despite their widespread use, a significant gap remains in the literature regarding the transition from laminar to turbulent flow within such geometries. Most existing studies rely primarily on pressure drop measurements for flow characterization, while comprehensive velocity-based investigations remain scarce. In this work, a detailed qualitative and quantitative characterization of the laminar–turbulent transition in a helical reactor is presented. An experimental set-up was developed to perform high-speed particle image velocimetry (PIV) measurements across a wide range of Reynolds numbers (Re = 500…9500). Various markers are employed to identify and analyse the onset of turbulence: visual flow patterns represented through carpet plots and pseudo-3D visualizations, statistical quantities such as standard deviation and turbulence intensity and the root mean square (RMS) of vorticity as a Lyne vortex indicator. Additionally, spectral analyses are conducted using power spectral density (PSD) evaluation, along with the estimation of dissipation rates and Kolmogorov time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tau$</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> and length scales <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\eta}_{\tau }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>η</mi> <mi>τ</mi> </msub> </math></EquationSource> </InlineEquation> to assess the small-scale turbulent structures. Pseudo-3D visualizations reveal first turbulent structures at <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>2400</mn> </mrow> </math></EquationSource> </InlineEquation> and a constant presence of additional vortex structures at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=6000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>6000</mn> </mrow> </math></EquationSource> </InlineEquation>. Analysis of the standard deviation and turbulence intensity reveals a transition between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400\dots 4800$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>2400</mn> <mo>⋯</mo> <mn>4800</mn> </mrow> </math></EquationSource> </InlineEquation>. The frequency analysis with PSD shows full turbulence at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=7000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>7000</mn> </mrow> </math></EquationSource> </InlineEquation>. Vorticity analysis and dissipation rates support a transition area between <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400\dots 5000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>2400</mn> <mo>⋯</mo> <mn>5000</mn> </mrow> </math></EquationSource> </InlineEquation>. These various ranges reveal that predicting exactly the transition point of course depends on the retained definition, with many different approaches documented in the literature. Keeping this in mind, it is still possible to give a lower boundary for the onset of transition around <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>2400</mn> </mrow> </math></EquationSource> </InlineEquation> and an upper boundary to a fully turbulent flow within the whole cross section at <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=7000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Re</mtext> <mtext>crit</mtext> </msub> <mo>=</mo> <mn>7000</mn> </mrow> </math></EquationSource> </InlineEquation> for the considered geometry (curvature ratio <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta =0.847$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.847</mn> </mrow> </math></EquationSource> </InlineEquation>, inner tube diameter <InlineEquation ID="IEq11"> <EquationSource Format="TEX">${d}_{i}=10 \text{m}\text{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>10</mn> <mtext>mm</mtext> </mrow> </math></EquationSource> </InlineEquation>).</p> Graphical abstract <p></p>

Characterization of the laminar–turbulent transition in helically coiled reactors: an experimental study using high-speed PIV and LDV

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2026年7月10日

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Conrad Müller,
Péter Kováts,
Dominique Thévenin,
Katharina Zähringer

摘要

Helically coiled reactors (HCRs) are increasingly employed in industrial applications due to their enhanced heat and mass transfer characteristics, compact design and improved mixing performance compared to straight tube systems. Despite their widespread use, a significant gap remains in the literature regarding the transition from laminar to turbulent flow within such geometries. Most existing studies rely primarily on pressure drop measurements for flow characterization, while comprehensive velocity-based investigations remain scarce. In this work, a detailed qualitative and quantitative characterization of the laminar–turbulent transition in a helical reactor is presented. An experimental set-up was developed to perform high-speed particle image velocimetry (PIV) measurements across a wide range of Reynolds numbers (Re = 500…9500). Various markers are employed to identify and analyse the onset of turbulence: visual flow patterns represented through carpet plots and pseudo-3D visualizations, statistical quantities such as standard deviation and turbulence intensity and the root mean square (RMS) of vorticity as a Lyne vortex indicator. Additionally, spectral analyses are conducted using power spectral density (PSD) evaluation, along with the estimation of dissipation rates and Kolmogorov time $\tau$ $τ$ and length scales ${\eta}_{\tau }$ $η_{τ}$ to assess the small-scale turbulent structures. Pseudo-3D visualizations reveal first turbulent structures at ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400$ ${Re}_{crit} = 2400$ and a constant presence of additional vortex structures at ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=6000$ ${Re}_{crit} = 6000$ . Analysis of the standard deviation and turbulence intensity reveals a transition between ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400\dots 4800$ ${Re}_{crit} = 2400 \dots 4800$ . The frequency analysis with PSD shows full turbulence at ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=7000$ ${Re}_{crit} = 7000$ . Vorticity analysis and dissipation rates support a transition area between ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400\dots 5000$ ${Re}_{crit} = 2400 \dots 5000$ . These various ranges reveal that predicting exactly the transition point of course depends on the retained definition, with many different approaches documented in the literature. Keeping this in mind, it is still possible to give a lower boundary for the onset of transition around ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=2400$ ${Re}_{crit} = 2400$ and an upper boundary to a fully turbulent flow within the whole cross section at ${\text{R}\text{e}}_{\text{c}\text{r}\text{i}\text{t}}=7000$ ${Re}_{crit} = 7000$ for the considered geometry (curvature ratio $\delta =0.847$ $δ = 0.847$ , inner tube diameter ${d}_{i}=10 \text{m}\text{m}$ $d_{i} = 10 mm$ ).

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