---
title: 'CLVisc: 3+1D Relativistic Hydrodynamic Framework'
url: https://www.emergentmind.com/topics/clvisc-hydrodynamic-framework
type: topic
---

# CLVisc: 3+1D Relativistic Hydrodynamic Framework

CLVisc is a \((3+1)\)-dimensional relativistic hydrodynamic framework for heavy-ion collisions that serves as the spacetime evolution engine between fluctuating initial conditions and particlization. In the literature provided here, it appears as an event-by-event viscous hydrodynamic code, a GPU/OpenCL implementation of \((3+1)\)D hydrodynamics, a finite-net-baryon-density extension with baryon diffusion, and the hydrodynamic backbone for calculations of soft hadrons, heavy flavor, directed flow, and hyperon spin polarization [1802.04449, 2107.04949].

## 1. Historical development and scientific role

The framework is explicitly identified in the 2018 implementation paper as a \((3+1)\)D viscous hydrodynamic model parallelized on GPUs using OpenCL, with “60 times performance increase for space-time evolution and more than 120 times for the Cooper-Frye particlization relative to that without GPU parallelization” [1802.04449]. In that formulation, CLVisc is already positioned as an event-by-event tool for observables that depend on longitudinal fluctuations, such as \(v_n(\eta)\) and longitudinal flow decorrelation.

A major later step is the finite-density extension to RHIC Beam Energy Scan conditions. That version adds net baryon number conservation, an Israel–Stewart-like equation for baryon diffusion, the NEOS-BQS equation of state, fluctuating \(3\)D Monte-Carlo Glauber initial conditions, and the SMASH afterburner, and is applied across \(\sqrt{s_{NN}}=7.7\)–\(62.4\) GeV [2107.04949]. This extension changes CLVisc from a baryon-free high-energy framework into a finite-\(\mu_B\) baseline for BES phenomenology.

Across subsequent applications, CLVisc is used as the intermediate dynamical stage in multistage chains rather than as a standalone model. The cited papers couple it to AMPT for local spin polarization [2002.03360], to TRENTo for hydrodynamic \(D\)-meson production [2101.02356], to tilted optical Glauber initial conditions for directed flow [2208.00155], to SMASH or AMPT initial conditions for helicity polarization at BES energies [2304.08777], to TRENTo-3D or TRENTo3D for left-right splitting and polarization studies [2505.14637, 2606.09093], and to KoMPoST-3D in a McDIPPER+KoMPoST-3D+CLVisc+SMASH chain for pre-equilibrium-to-hydrodynamics matching [2512.08007]. This corpus consistently presents CLVisc as the hydrodynamic response module that converts initial geometry, flow, and conserved-charge structure into freeze-out hypersurfaces and local gradients.

## 2. Hydrodynamic structure and evolved fields

In the general viscous formulation used in the CLVisc literature, hydrodynamic evolution is built on local conservation laws. Several papers write the system as
\[
\nabla_\mu T^{\mu\nu}=0,
\qquad
\nabla_\mu J^\mu=0,
\]
with a dissipative decomposition of the energy-momentum tensor and current [2107.04949, 2606.09093]. A finite-density version is written as
\[
T^{\mu\nu}=eU^\mu U^\nu-P\Delta^{\mu\nu}+\pi^{\mu\nu},
\qquad
J^\mu=nU^\mu+V^\mu,
\qquad
\Delta^{\mu\nu}=g^{\mu\nu}-U^\mu U^\nu,
\]
while other CLVisc applications include bulk pressure explicitly through
\[
T^{\mu\nu}=\varepsilon u^\mu u^\nu-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}.
\]
This is not a contradiction; it reflects paper-dependent choices about whether bulk viscosity or conserved charges are retained in a given application [2208.00155, 2606.09093].

The 2018 implementation paper states that CLVisc works in the Landau frame and uses Milne coordinates \(X^\mu=(\tau,x,y,\eta_s)\), with the Milne metric
\[
g^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1/\tau^2).
\]
That paper also gives the projector orthogonal to the flow,
\[
\Delta^{\mu\nu}=g^{\mu\nu}-u^\mu u^\nu,
\]
and formulates the second-order viscous sector in an Israel–Stewart-type structure with shear-stress transversality and tracelessness [1802.04449].

For BES applications, CLVisc evolves baryon diffusion with an Israel–Stewart-like relaxation equation,
\[
\Delta^{\mu\nu}DV_{\nu}
=
-\frac{1}{\tau_V}\left(V^\mu-\kappa_B\nabla^\mu\frac{\mu_B}{T}\right)
-
V^\mu\theta
-
\frac{3}{10}V_\nu\sigma^{\mu\nu},
\]
together with a diffusion coefficient parameterized as
\[
\kappa_B=\frac{C_B}{T}n\left(\frac{1}{3}\cot\left(\frac{\mu_B}{T}\right)-\frac{nT}{e+P}\right).
\]
The same finite-density paper gives
\[
\tau_\pi=\frac{5C_{\eta_v}}{T},
\qquad
\tau_V=\frac{C_B}{T},
\]
and uses constant \(C_{\eta_v}=0.08\) and \(C_B=0.4\) across BES energies [2107.04949].

What CLVisc supplies to downstream analyses depends on the application, but the recurring output fields are the local temperature, flow velocity, conserved-charge information when present, dissipative tensors, and the freeze-out hypersurface. In the BES helicity-polarization study, the downstream calculation explicitly uses \(T(X)\), \(\mu(X)\), and \(u^\mu(X)\) on the chemical freeze-out hypersurface, together with structures such as
\[
\boldsymbol{\omega}=\nabla\times\mathbf{u},
\qquad
\pi_{\sigma j}
=
\partial_\sigma u_j+\partial_j u_\sigma-u_\sigma(u\cdot\partial)u_j,
\]
to decompose helicity polarization into thermal, shear, acceleration/temperature-gradient, and chemical components [2304.08777].

## 3. Numerical implementation, coordinates, and particlization

The explicit implementation paper makes clear that CLVisc is not only a hydrodynamic formalism but also a computational framework. It uses the second-order central Kurganov–Tadmor scheme for the convective part of the conservation laws and a two-step Runge–Kutta predictor-corrector update for the coupled viscous evolution [1802.04449]. The same finite-density extension states that the hydrodynamic equations are solved with the Kurganov–Tadmor finite-volume algorithm and a second-order Runge–Kutta time integrator, again emphasizing low numerical viscosity and flux-based evolution [2107.04949].

GPU acceleration is a defining feature of the original implementation. The framework is parallelized with OpenCL, and the paper details acceleration of the hydrodynamic evolution, freeze-out hypersurface finding, and Cooper–Frye integration [1802.04449]. A representative later performance example reports that, for a grid size
\[
N_x\times N_y\times N_\eta = 201\times 201\times 201,
\]
a smooth-initial-condition \(0\)–\(5\%\) Au+Au event at \(\sqrt{s_{NN}}=19.6\) GeV on an NVIDIA GeForce GTX 3080 takes about \(\sim 1800\) s for the hydro event total, \(\sim 1.6\) s per time step, \(\sim 0.066\) s per particle sample event, and \(\sim 2.5\) s per SMASH afterburner event [2107.04949].

Particlization is consistently performed with Cooper–Frye. In the finite-density formulation this is written as
\[
\frac{dN}{dY\,p_Tdp_Td\phi}
=
\frac{g_i}{(2\pi)^3}
\int_\Sigma p^\mu d\Sigma_\mu\,
f_{\rm eq}(1+\delta f_\pi+\delta f_V),
\]
with the switching hypersurface found using the Cornelius routine [2107.04949]. In the pre-equilibrium matching study, the CLVisc-to-SMASH handoff is specified at
\[
T_{\rm frz}=0.154\ \mathrm{GeV},
\]
again through Cooper–Frye with a shear viscous correction \(\delta f_\pi\) [2512.08007]. Other studies adopt constant-energy-density or constant-temperature criteria, for example
\[
e_{\rm frz}=0.4~\mathrm{GeV/fm^3}
\]
in several RHIC applications [2107.04949, 2505.14637, 2606.09093].

The implementation literature also shows that CLVisc is designed to support both smooth spectra integration and Monte Carlo sampling. The 2018 paper tabulates smooth spectra on
\[
N_Y\times N_{pt}\times N_\phi = 41\times 15\times 48
\]
bins and describes Monte Carlo sampling that can interface with hadronic transport codes such as UrQMD, JAM, or SMASH [1802.04449]. In practice, some studies use SMASH as an actual afterburner, whereas others include resonance decays but no hadronic rescattering [2107.04949, 2505.14637].

## 4. Initialization strategies, equations of state, and finite-density generalization

A central feature of CLVisc is its adaptability to multiple initial-condition models. The cited studies use optical Glauber [1906.09385, 2208.00155], Monte-Carlo Glauber [2107.04949], AMPT [1802.04449, 2002.03360], TRENTo [2101.02356], TRENTo-3D or TRENTo3D [2408.04296, 2505.14637, 2606.09093], SMASH initial conditions at BES energies [2304.08777], and a full pre-equilibrium energy-momentum tensor from KoMPoST-3D [2512.08007]. This variety is important: CLVisc is not tied to one initialization paradigm.

The same diversity appears in the equation of state. The implementation paper lists EOSI, lattice-wb2014, s95p-pce, EOSQ, and pure gauge as available choices, with s95p-pce used as the default in much of that paper’s phenomenology [1802.04449]. The charm-hydrodynamics study uses a partial chemical equilibrium EoS with a chemical freeze-out temperature of \(165\) MeV [2101.02356]. The directed-flow paper uses the Wuppertal-Budapest 2014 lattice QCD EoS [2208.00155]. Several recent top-RHIC and LHC applications use HotQCD-2014 or HotQCD-based equations of state [2505.14637, 2606.09093]. At BES energies, the finite-density extension uses NEOS-BQS, which is designed for matter with conserved baryon, electric charge, and strangeness content [2107.04949]. The helicity-polarization study specifically selects NEOS-BQS because RHIC-BES matter is neither boost-invariant nor baryon-free [2304.08777].

The finite-density extension is the clearest statement of CLVisc as a conserved-charge hydrodynamic framework. It initializes both entropy density and net baryon density in \(x,y,\eta_s\),
\[
s(x,y,\eta_s)=\frac{K}{\tau_0}\left(H_P^s(\eta_s)s_P(x,y)+H_T^s(\eta_s)s_T(x,y)\right),
\]
\[
n(x,y,\eta_s)=\frac{1}{\tau_0}\left(H_P^n(\eta_s)s_P(x,y)+H_T^n(\eta_s)s_T(x,y)\right),
\]
with \(\sigma_r=0.5\) fm and energy-dependent \(\tau_0\) chosen larger than the nuclear overlap time at low beam energy [2107.04949]. By contrast, several top-energy studies explicitly set \(n=0\) and ignore baryon diffusion because midrapidity is approximately baryon-free in those settings [2505.14637, 2606.09093].

Recent TRENTo3D applications show another direction of framework evolution: CLVisc can be initialized not only with scalar energy-density profiles but also with structured longitudinal momentum components. In the isobaric Zr+Zr polarization study, TRENTo3D is modified through
\[
T^{\tau\tau}=\varepsilon_{\rm IC}(\vec{x}_\perp,\eta_s)\cosh(y_L),
\qquad
T^{\tau\eta_s}=\frac{1}{\tau_0}\varepsilon_{\rm IC}(\vec{x}_\perp,\eta_s)\sinh(y_L),
\qquad
y_L=f_v\cdot\eta_{\rm cm}(\vec{x}_\perp),
\]
so that an initial longitudinal flow gradient seeds vorticity before CLVisc evolution [2606.09093]. The KoMPoST-3D coupling goes further by passing a full out-of-equilibrium \(T^{\mu\nu}(\tau_{\rm hydro},x,y,\eta_s)\) into Landau-frame decomposition before CLVisc evolution [2512.08007].

A practical implication, directly supported by the cited applications, is that there is no single universal CLVisc parameter set. Some papers give explicit values of \(\tau_0\), \(\eta/s\), \(T_{\rm frz}\), or \(e_{\rm frz}\), while others state only that parameters are adopted from earlier CLVisc studies or tuned to multiplicity and elliptic flow [2002.03360, 2509.00380]. This suggests that “CLVisc” names a hydrodynamic infrastructure rather than a unique phenomenological calibration.

## 5. Observable classes and phenomenological reach

CLVisc’s original benchmark domain is soft-hadron bulk phenomenology. The 2018 paper uses AMPT initial conditions with longitudinal fluctuations to compute \(v_n(\eta)\) and the decorrelation observable
\[
r_n(\eta^a,\eta^b)
=
\frac{\langle \vec{Q}_n(-\eta^a)\vec{Q}_n^*(\eta^b)\rangle}
{\langle \vec{Q}_n(\eta^a)\vec{Q}_n^*(\eta^b)\rangle},
\]
finding that the magnitude of \(v_n(\eta)\) depends on \(\eta_v/s\), but the shape of the \(v_n(\eta)\) distributions does not [1802.04449]. The BES finite-density extension broadens this to identified-particle spectra, mean transverse momenta, \(v_2\{2\}\), \(v_3\{2\}\), and \(v_2\{4\}/v_2\{2\}\) across \(7.7\)–\(62.4\) GeV, providing what that paper describes as a benchmark for RHIC-BES soft-hadron data [2107.04949].

The framework is also used for observables that require explicitly longitudinal and rapidity-odd dynamics. In the directed-flow study, CLVisc evolves a tilted optical-Glauber initial condition and traces the chain from rapidity-dependent tilt to asymmetric pressure gradients, rapidity-odd \(\langle v_x(\eta_s)\rangle\), and final hadron \(v_1\) in Au+Au, Ru+Ru, and Zr+Zr collisions at \(\sqrt{s_{NN}}=200\) GeV [2208.00155]. In the left-right elliptic-flow splitting analysis, TRENTo-3D+CLVisc is used to show that \(\Delta v_2(\eta)\) is mainly controlled by \(v_1\), while \(\Delta v_2(p_T)\) at midrapidity receives a substantial, even dominant, contribution from \(v_3^{\rm RP}\) [2505.14637].

Heavy flavor appears in a more restrictive role. The \(D^0\)-meson study uses event-by-event \((3+1)\)D viscous CLVisc as the hydrodynamic realization of an “extreme thermalization” scenario in which low-\(p_T\) charm quarks are assumed to comove with the fluid and hadronize statistically on the same freeze-out surface as light hadrons [2101.02356]. In that setup, CLVisc reproduces low-\(p_T\) \(D^0\) yields and much of the measured \(v_2\), but overpredicts the spectrum at higher \(p_T\), supporting the paper’s conclusion that only the lowest-\(p_T\) charm sector is close to local thermal equilibrium.

A particularly extensive application domain is spin polarization. In the early local-polarization work, CLVisc provides \(T(x)\), \(u^\mu(x)\), and freeze-out hypersurfaces from which several candidate vorticity tensors are constructed and inserted into a polarization formula. Those studies show that the predicted local polarization depends strongly on the choice of spin chemical potential, with temperature-vorticity-based choices giving qualitatively coincidental agreement with RHIC measurements for some observables [1906.09385, 2002.03360]. At BES energies, the helicity-polarization study uses relativistic \((3+1)\)D CLVisc hydrodynamics with SMASH initial conditions and finds that the kinetic-vorticity contribution
\[
P_H^\omega \propto \widehat{\mathbf p}\cdot(\nabla\times\mathbf u)
\]
dominates over shear, acceleration/temperature-gradient, and chemical-potential-gradient terms at intermediate and low collision energies, especially at \(\sqrt{s_{NN}}=7.7\) GeV [2304.08777].

Recent spin studies push CLVisc into more differential observables. In Au+Au at \(200\) GeV and p+Pb at \(8.16\) TeV, CLVisc is used to compute
\[
\left\langle P_z\sin 2(\phi_p-\Psi_2)\right\rangle
\]
under \(\Lambda\)-equilibrium, \(s\)-quark-equilibrium, and iso-thermal-equilibrium scenarios. The cited papers report that Au+Au data are described in the \(s\)-quark and iso-thermal scenarios, whereas p+Pb remains a puzzle because the hydrodynamic thermal-vorticity and shear-induced contributions do not reproduce the observed signal [2408.04296, 2509.00377]. In isobaric \(^{96}\mathrm{Zr}+^{96}\mathrm{Zr}\) collisions, TRENTo3D + CLVisc with an initial longitudinal flow gradient provides a simultaneous description of STAR measurements of global polarization \(-P^y\) and the azimuthal coefficients \(P_{y,\mathrm{c0}}\) and \(P_{y,\mathrm{c2}}\), while still overpredicting the high-\(p_T\) modulation amplitude of \(P_z\) [2606.09093].

## 6. Limits of the framework and current extensions

The cited literature repeatedly distinguishes CLVisc as a hydrodynamic backbone from the observable-specific modules layered on top of it. This distinction is especially clear in the spin sector. The BES helicity-polarization paper states that spin polarization is computed on top of the hydrodynamic background rather than evolved as an independent hydrodynamic degree of freedom; no explicit spin-hydrodynamic backreaction is implemented, and no separate spin diffusion or axial-charge evolution is solved [2304.08777]. This is important because it defines standard CLVisc polarization results as hydrodynamic baselines rather than complete spin-transport calculations.

The framework’s approximations are also strongly application dependent. Some studies use smooth optical-Glauber initial conditions and therefore address only rapidity-odd \(v_1\), not rapidity-even components or event-by-event fluctuation effects [2208.00155]. Several top-energy studies set \(n=0\), omit baryon diffusion, and exclude hadronic rescattering after freeze-out [2505.14637, 2606.09093]. The finite-density BES baseline itself neglects bulk viscosity and has no explicit pre-equilibrium stage, with the authors noting that these omissions matter most strongly at the lowest beam energies [2107.04949].

Small systems expose another limitation. In p+Pb polarization studies, the authors explicitly warn that the lowest-multiplicity results are unreliable because of the limitations of hydrodynamic applicability, and they conclude that current hydrodynamic effects alone cannot explain the observed beam-direction spin polarization [2408.04296, 2509.00380]. This suggests that CLVisc can successfully generate realistic bulk backgrounds in small systems only up to the point where additional non-hydrodynamic or non-flow spin mechanisms become important.

Current extensions in the literature move in three directions. One is pre-equilibrium matching: the McDIPPER+KoMPoST-3D+CLVisc+SMASH study uses CLVisc as the downstream viscous stage after a non-boost-invariant kinetic-theory response calculation and finds that final observables become only weakly sensitive to the hydrodynamic switching time \(\tau_{\rm hydro}\) over a broad range [2512.08007]. A second direction is non-extensive hydrodynamics: NEX-CLVisc is developed “in the framework of CLVisc where the viscous corrections are turned off,” modifying initial conditions, the equation of state, and kinetic freeze-out through Tsallis-like non-extensive effects [2408.12405]. A third direction is beyond-baseline spin physics: the recent local-polarization proceedings explicitly connect the present CLVisc-based calculations to interaction corrections from quantum kinetic theory and to spin hydrodynamics, while emphasizing that such corrections are not yet standard built-in components of CLVisc itself [2509.00377].

Taken together, these studies define CLVisc as a flexible \((3+1)\)D hydrodynamic infrastructure rather than a single fixed model. Its central identity is the evolution of realistic three-dimensional QCD matter, with viscous and, when needed, finite-density dynamics, from an externally specified initial state to a freeze-out hypersurface. Its scientific value lies in the fact that the same infrastructure can be coupled to very different initial-state models, equations of state, and observable modules; its limitations arise precisely where hydrodynamics, local equilibrium, or freeze-out-based post-processing become insufficient.

Source: https://www.emergentmind.com/topics/clvisc-hydrodynamic-framework